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PHYS 102 Lab Manual 2007 Edition Department of Physics King Fahd University of Petroleum & Minerals Dhahran 31261 Saudi Arabia King Fahd University of Petroleum & Minerals PHYS102 TABLE OF CONTENTS Title of Experiment Page Preface iv Laboratory Policy v The Skills You Should Learn from Doing Experiments vi Data Analysis and Presentation 1 Standing Waves on a String 5 Speed of Sound in Air 8 Standing waves in a Cylindrical Tube 12 Specific Heat of a Metal 16 Perfect Gas Law 19 Ohm's Law-1 22 Ohm’s Law-2 25 Wheatstone Bridge 28 From Galvanometer to Voltmeter 32 Slide-Wire Potentiometer 37 Capacitors in Series and Parallel 40 Introduction to the CRO 44 Measurement of e/m of the Electron 50 Tangent Galvanometer 55 Internal Resistance and EMF of a Battery 58 Kirchhoff's Laws 61 Introduction to Electrical Measurements 64 Measurement of Thickness of a Very Thin Sample 65 © KFUPM – PHYSICS revised 29/11/2007 ii Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Cal Lab-Waves in Excel 68 Appendix A 78 © KFUPM – PHYSICS revised 29/11/2007 The Laboratory Barometer iii Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 PREFACE This manual contains laboratory experiments for “General Physics II” (Physics 102). These experiments have been designed to acquaint freshman students with the fundamentals of apparatus manipulation, physical measurements, data recording and analysis aimed at verifying known laws. It is hoped that comments and suggestions from both students and instructors will help bring up new experiments and improve existing ones. Physics Department Dhahran 2003 © KFUPM – PHYSICS revised 29/11/2007 iv Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 LABORATORY POLICY Supplies Students must bring the supplies they need, such as laboratory notebook (graph notebook), pencils, ruler, eraser and calculator, as none of these items will be provided to them in the lab. Attendance 1. Attending the lab session is compulsory. 2. If and when the total number of unexcused absences reaches three (3) or unexcused and/or excused absences reaches five (5), a grade of "DN" will be assigned for the course. 3. A student absent from a lab session and who submits an official excuse will be allowed to make up the associated experiment, if possible. The case of a student who has no official excuse will be dealt with at the discretion of the instructor. 4. A student cannot make up a lab with another section without a written request to that effect from his own lab instructor. Lab Grade 1. The lab work comprises 20% of the total score for the course. The final lab grade will be calculated according to the prevailing policy. 2. The lab grade will be based on reports, final lab exam and quizzes, the latter at the discretion of the instructor. The final lab exam is compulsory, and it may either be written or practical or both, at the discretion of the instructor. Preparation, Lab work, etc..... 1. Students should read the write-up of each experiment before coming to the lab. 2. All experiments have been designed so that they can be completed within the allotted time of 3 hours. 3. Students should arrive in time for their lab. Late arrival will be dealt with at the discretion of the instructor. 4. A student found in possession of an old lab report during a lab session will get a zero for that lab irrespective of whether he used that lab report or not. 5. Students are required to leave the equipment in a proper state after they finish an experiment. Electrical appliances, if used, should be switched off and disconnected. © KFUPM – PHYSICS revised 29/11/2007 v Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 THE SKILLS YOU SHOULD LEARN FROM DOING EXPERIMENTS You are doing labs to demonstrate and/or verify the laws of physics. Having passed Physics 101 and done the labs, we hope you have acquired the skills, which we mentioned in the 101 Lab manual. In 102 Labs, we continue to emphasise the same skills, which we hope will remain with you well after you have finished with the general physics courses and will be helpful in your future careers. For this reason your instructor will constantly emphasise on these skills. In particular, the lab final exam may test if you have learnt some of these skills. The four types of skills which we hope you will acquire are: 1. Experimental Skills. • • • • • 2. Record all measurements taken. Repetitive measurements should be tabulated. Know that every measurement is subject to uncertainty (“error”). Know how to estimate error in each measurement and thus be able to identify the major sources of errors. Emphasise units and significant figures. Be familiar with measuring instruments used in the experiments and to choose the appropriate scale for more precise readings. Know how to follow experimental procedures e.g. connecting circuits, taking data. Graphical Skills. • • 3. How to linearise the equation so as to plot a straight line graph. How to find the slope/intercepts from the graph and relate them to the linearised equation. Use of proper scales in plotting graphs. Label the axes. Show units. Analytical Skills. • • 4. To infer relationships, if any, between sets of data. Draw conclusions from experimental results and compare with theory. Common Sense Skills. • Check your result to see if it makes sense. You should be able to judge whether a measured/calculated value is reasonable or not. Example: a current of a few millions of amperes reported in the labs is definitely not a reasonable result! © KFUPM – PHYSICS revised 29/11/2007 vi Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 DATA ANALYSIS AND PRESENTATION Purpose To learn how to analyse experimental data and to practise error analysis. Exercise (1) The data in the following table relates to measurements made of the period of oscillation T of a simple pendulum of variable length L. Theory predicts the following relationship: T=2π L g (1) where g is the acceleration due to gravity. Table 1 Length (L) cm ± 0.5 cm 57.8 47.5 41.5 35.0 29.0 22.5 16.5 Period (T) sec ± 0.05 sec 1.50 1.41 1.26 1.19 1.08 0.93 0.80 g (cm/s2) 1. Copy Table 1 and calculate a value of g for each data point using equation (1). Find the average of g for all data points. 2. Write down the general expression for the fractional maximum possible error (MPE) (relative uncertainty in g) ∆ g/g in terms of ∆ T/T and ∆ L/L (Read the Appendix). 3. Calculate ∆ g for one of the data point, say for L = 41.5 cm, from the table in step (1). 4. Look at the g values in Table 1 and make another estimate of the MPE of g (call it ∆ g ' ) from the range or spread of the data, i.e. (∆g')range = 5. gmax - gmin 2 Using the data supplied construct another table as follows: © KFUPM – PHYSICS revised 29/11/2007 1 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Table (2) Length (L) Period (T) cm sec - T2 sec2 - - ± ∆L ± ∆T cm sec - - ± ∆ (T2 ) = 2T∆ T sec2 - Use the data in this table to plot a best-fit straight-line graph and determine g from the slope. Include error bars and determine ∆ g from your graph. Hint: To find ∆ gslope , find the maximum and minimum slopes and calculate the corresponding values of g. ∆ gslope = maximum g - minimum g 2 6. Compare the different estimates of the MPE of g in steps (3), (4) and (5). 7. Prove that ∆(T 2 ) = 2T∆T . Exercise (2) Two lengths are measured with a meter ruler X = 35.5 cm and Y = 67.3 cm, where the uncertainty is ∆X = ∆Y = 0.2 cm . 1. Find the percentage error in Z1, where Z1 = X + Y. 2. Find the percentage error in Z2, where Z2 = X – Y. 3. Find the percentage error in Z3, where Z3 = X . Y 4. Find the percentage error in Z4, where Z4 = XY. 5. Find the percentage error in Z5, where Z5 = X 2 /Y3 . © KFUPM – PHYSICS revised 29/11/2007 2 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 APPENDIX 1. Simple Error Analysis Suppose that a quantity Xo is measured. It can usually be stated that the uncertainty in the measurement lies within some 'reasonable maximum range', i.e. Xo ± ∆ X where ∆X is called the 'maximum possible error' MPE (uncertainty in X o ), ∆X is Xo called the fractional error (relative error) in Xo and {(∆ X ) X 100 } is the percentage Xo error. 2. Error Calculations and Differential Calculus In the past you used to express errors as percentage errors when dealing with experimental measurement. In general, now that you have learned the elements of calculus, you will find that it is much easier to derive results by differentiation. We will show this by examples. Example (1) : Simple Algebraic Relation Suppose Y = X2 where X is measured in an experiment and Y is calculated. Suppose there is an error ∆X associated with the observed value X1. Then there is a corresponding error ∆Y in the calculated value of Y (see diagram). If ∆ X (and ∆ Y) are small enough, we can treat them as differential quantities. (This assumes that Y is reasonably smooth and continuous function, which is always true in experimental Physics). © KFUPM – PHYSICS revised 29/11/2007 3 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 By differentiation we obtain dY = 2X dX When the changes are definite, we can write ∆ Y = 2X ∆ X ∆Y = 2X ∆X = 2 ∆X = 2 (fractional error in X) X Y X2 Example (2) : General Algebraic Relation Let Y = A Xn where A is a constant and n can be negative or a fraction. Differentiating dY = A n Xn-1 dX ∆Y = A n Xn-1 ∆X ∆Y = A n X n-1 ∆X = n { ∆X } X Y A Xn Fractional error in Y = | n | (Fractional error in X) Example (3): Functions of Several Variables Consider the case where Z is a function of two variables A and B, i.e. Z = Z (A, B). The expressions for ∆Z for some common relations between Z and A, B are given in the following table: Relation between Z and A, B Relation between errors Z=A+B ∆ Z = ∆A + ∆B Z=A-B ∆ Z = ∆A + ∆B Z A B Z = AB Z = A/B © KFUPM – PHYSICS revised 29/11/2007 4 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 STANDING WAVES ON A STRING Purpose To investigate standing waves in a vibrating string. To measure the frequency of a vibrator. Introduction When two equal wave trains traveling in opposite directions act upon a series of particles, the resulting phenomenon is called stationary or standing waves. This type of wave motion may be produced by either longitudinal or transverse waves. Standing waves on mechanical structures are of basic importance to engineering design, and standing waves on strings is the physical basis of all stringed musical instruments. Take a string stretched between fixed supports in a system that can carry a wave, but which has definite 'boundary conditions' (e.g. the ends of the string at the support can not move). When the string is vibrated at a certain frequency, the amplitude developed at a given point is rather small because the multiple-reflections of the waves between the fixed ends lead to displacements that cancel out on average. Only at the resonant condition does the amplitude build up, because then the reflected waves are always reinforcing the incident waves. Consider a string under tension T and supported at two points. (See Fig. 1). When the string is set into vibration these two supporting points are forced to stay at rest. The traveling waves in the string are reflected at the supports, and they combine to form a stationary pattern, the so-called standing wave. The fundamental mode of vibration is shown in Fig. 1 with the distance between the supports equal to λ /2 (λ is the wavelength of the traveling waves). The wave created travels along the wire with a speed "V" given by the equation V = T (1) µ where T is the tension in the string and µ is the mass per unit length of the string. Figure 1 © KFUPM – PHYSICS revised 29/11/2007 5 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 It is a well established fact that the wavelength λ , in any type of wave motion, is related to the frequency, f, and the velocity, V, by the relation V = fλ (2) This standing wave pattern will occur only if the tension and length of the string are properly adjusted. These two oppositely directed wave trains with the same frequency are superimposed upon the string in such a way as to give alternate regions of no vibration N, i.e., minimum displacement, and regions of maximum vibrations A, i.e. maximum displacement. The regions N and A are called nodes and antinodes, respectively, and the segment between two nodes is called a loop. Figure 2 shows the first three modes of the standing wave pattern. Figure 2. λ1 ; (fundamental mode) ; hence λ1 = 2 L / 1 (a) L= (b) L = λ 2 (first overtone mode) ; hence λ 2 = 2 L / 2 (c) L= 2 3λ3 (Second overtone mode) ; hence λ3 = 2 L / 3 2 © KFUPM – PHYSICS revised 29/11/2007 6 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 In general, for an n-loop standing wave, λn = 2L / n Combining equations (1), (2) and (3), we get (3) n2 = 4L2 f2 µ ( 1 ) (4) T In this experiment, a string of length, L, is forced to vibrate at the frequency, f, of a vibrator. Changing the tension in the vibrating string causes a change in the number of loops, n, between the ends of the string. Data 1. Using the sample string provided, find its length, L', and mass, m. 2. Plug in the vibrator and add masses on the hanger until a two-loop (n = 2) standing wave is established.* Record the total mass M suspended. 3. Gradually decrease the mass suspended until a three-loop (n = 3) standing wave is observed. Record the total mass M. 4. Continue decreasing the mass until standing waves with n = 4, n = 5 and n = 6 loops are observed. In each case record the values of M. Tabulate your measurements. 5. Measure the length, L, of the vibrating string. Analysis 1. Calculate the linear mass density of the string, µ = m / L' . 2. Use equation (4) and the data collected to plot a straight line graph. 3. Calculate the frequency of the vibrator, f, from the slope of the graph. Exercises 1. What is the major source of error in this experiment? 2. All strings on a violin are of the same length. What difference do they have that gives them different frequencies (different pitch)? 3. A string 4.0 m long has a mass of 0.40 gram and is driven by a source at a frequency of 80 Hz. Find the mass of the weight that must be hung from the string if it is to resonate in four segments. 4. Two strings connected end to end are of lengths L1 and L2 and of masses per unit length of µ1 and µ2 . How should the ratio L1 / L2 be chosen so that the strings will resonate with a node at their junction when a wave is sent along them? Assume n1 = n2. (In practice, a wave will also be reflected from their junction. Assume this effect to be negligible). ___________________________________________________________________________ * No need to find the fundamental mode of vibration. This would require a very large mass which would cause the string to break. © KFUPM – PHYSICS revised 29/11/2007 7 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 SPEED OF SOUND IN AIR Purpose To study sound-wave resonances in air columns and to measure the velocity of sound in air. Introduction For a periodic wave the frequency f, the wavelength λ , and the velocity v are related by the following equation: v = fλ (1) The experimental determination of λ involves the production of standing waves. When a train of waves from a source is reflected at a boundary, a standing wave is set up. As an example figure 1 shows standing waves in air columns. When an organ pipe is used as a “resonance” chamber, standing waves are set-up in the air column in the pipe. The simplest resonance chamber consists of a tube open at one end and closed at the other, the open end being near the sounding body. If the tube is of the proper length, resonant standing waves are set up which reinforce the sound emitted by the source. In such a resonator there is a node at the closed end, since no vibration can occur there, and an antinode at the open end for there the particles of air have the greatest freedom of motion. Thus the shortest closed pipe which will resonate with a source of a given frequency (i.e. with a given wave length λ ) is one whose length is one quarter wave length. This situation is illustrated in figure 1a. A pipe three times as long as the minimum will also resonate with the same note, as shown in figure 1b. A little thought will show that any closed pipe whose length is an odd number of multiples of a quarter wavelength will produce resonance. © KFUPM – PHYSICS revised 29/11/2007 8 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Owing to the fact that the maximum disturbance does not occur exactly at the mouth of the pipe, the distance A1 N1 (figure 1) is not exactly one quarter wave length. An end correction must be added to the measured length. The necessity for making this correction may be eliminated by using a long tube and measuring the successive internodal distances N1 N2, N2 N3 etc. The velocity of sound in an ideal gas is related to the properties of the medium by the equation v= γ P (2) ρ where P = pressure ρ = density and γ = specific heat ratio = Cp Cv = 1.403 for air. Use the Appendix B to find the value of ρ . Equation (2) may be rewritten in the form RT M where M = molecular weight of the gas T = temperature (K) and R = universal gas constant = 8.314 J/K.mole v= γ (3) Equation (3) shows that the velocity of sound in an ideal gas depends only on the temperature and not on the pressure or density of the gas. The temperature dependence of the sound velocity in an ideal gas may be written. vT = v273K T 273K (4) The value of v273K = 331 m/s Apparatus The resonance apparatus represented diagramatically in figure 2 consists of a glass resonance tube R supported vertically on an iron stand. The water level in the resonance tube is adjusted by regulating the height of the reservoir V which slides upon the iron standard and is connected to the resonance tube by a rubber hose. A metric scale S attached to the resonance tube, or mounted beside it, serves to measure the water level. A small loud speaker, P, connected to a signal generator, SG, is placed over the © KFUPM – PHYSICS revised 29/11/2007 9 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 mouth of the tube and is used to drive the air column to resonance. A tuning fork can be used instead of speaker and SG for this purpose. Procedure 1. Connect the small loud speaker to the signal generator and set the sine wave amplitude control to maximum. Choose a frequency in the range 250 to 500 Hz. With the reservoir V at its highest position, fill the resonance tube R nearly full of water. Hold the small loud speaker P over the mouth of the tube. (Remember tuning forks can also be used instead of speaker and SG.) 2. Slowly lower the reservoir and listen for an intensification of the sound as the length of the air column is increased. Locate the position of maximum intensity as closely as possible by raising and lowering the water level several times. Record the position of resonance. 3. Again lower the reservoir and locate and record the second resonance position. Find the internodal distance and calculate the wavelength and velocity of sound. 4. Repeat the experiment at two other frequencies and calculate the velocity of sound from the internodal distance in each case. Tabulate your measurements. 5. Observe and record the room temperature and pressure. © KFUPM – PHYSICS revised 29/11/2007 10 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Data Analysis 1. From all the measurements in step 3 and 4, determine the average experimental sound velocity. 2. Calculate the sound velocity using equation (2), and the handbook value for the density of air at the existing temperature and pressure. Convert pressure in cm of Hg to N/m2 using 76 cm Hg = 1.013 x 105 N/m2. 3. Calculate v using equation 4, and the existing room temperature. This value of v should, of course, be the same as that calculated from equation (2). 4. If the uncertainty in locating the position of resonance is, say, 1 cm, find the uncertainty in the measured speed of sound in steps 1 to 4 of Experimental Procedure. 5. Does the average velocity of sound obtained in step 1 agree, within experimental error, with the velocity calculated in steps 2 and 3? Exercises 1. Starting from equation (2), derive equation (3). 2. A glass tube (open at both ends) of variable length L is positioned near a loud speaker of frequency 680 Hz. Find the three smallest values of L for which the tube will resonate with the speaker (Use vsound = 343 m/s). © KFUPM – PHYSICS revised 29/11/2007 11 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 STANDING WAVES IN A CYLINDRICAL TUBE Purpose To study sound-wave resonances in a cylindrical tube and to measure the velocity of sound in air. Introduction For a periodic wave the frequency f, the wavelength λ , and the velocity v are related by the following equation: v=fλ (1) In this experiment, the apparatus shown in figure 1 is used to produce standing waves in air. A sound source is located at one end of the closed tube. A microphone attached to a stainless steel rod can be moved along the length of the tube to monitor the acoustic (sound) pressure. sound source gas port microphone gas port removable end plug to oscillator to oscilloscope Figure 1. The standing wave sound tube The sound source produces a sound wave that travels down the tube and is reflected from the other end of the tube. The incident and reflected waves interfere with each other, and at certain frequencies produce standing waves. If the tube, length L, is closed at both ends, the standing wave produced will have a displacement node (a position of no vibration) at each end. However, the pressure amplitude, being 90o out of phase with the displacement wave, will be maximum at each end. This is shown diagrammatically in Figure 2 for the first (fundamental) and second harmonics. L = λ/2 ; n =1 L=λ;n=2 Displacement amplitude Pressure amplitude Figure 2. Standing waves in a tube closed at both ends © KFUPM – PHYSICS revised 29/11/2007 12 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Therefore, resonance occurs at frequencies fn where the length of the tube, L, is an integer multiple of half wavelengths; that is L = n (λ/2), (n = 1, 2, 3, … ) and fn = n (v/2L) (2) where v is the velocity of sound. In part A of this experiment we shall measure the velocity of sound wave by finding the resonant frequencies when the tube is closed at both ends. In part B, we shall plot the pressure amplitude as a function of position from the sound source when the standing wave is in the fundamental mode, and obtain a value for v. For comparison, we shall calculate the velocity of sound in air (treated as an ideal gas) from the properties of the medium (air) given by the equation v = γ P (3) ρ where P = pressure ρ = density and γ = specific heat ratio = Cp = 1.403 for air. Cv Equation (3) may be rewritten in the form v= γ RT M (4) where M = molecular weight of the gas T = absolute temperature (K) and R = universal gas constant = 8.314 J/K.mole Equation (4) shows that the velocity of sound in an ideal gas depends only on the temperature and not on the pressure or density of the gas. The temperature dependence of the sound velocity in an ideal gas may thus be written vT = v273 T 273 (5) The value of v273 = 331 m/s © KFUPM – PHYSICS revised 29/11/2007 13 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Part A. Resonant frequencies in a tube closed at both ends. Procedure 1. Set the oscillator sine wave output voltage to zero and then connect it to the sound source (small loudspeaker) on the tube. The output of the oscillator is also connected to one vertical input of a dual-trace oscilloscope. Set the frequency to a few kHz and gradually increase the oscillator output to one volt peak to peak. [Do not exceed 2 volts peak to peak or damage to the sound source may occur.] Pull the microphone all the way until its front surface is level with the end plug. Connect the microphone leads to the second vertical input of the dual-trace oscilloscope, and switch on the microphone. 2. Set the frequency of the oscillator at 200 Hz. Slowly increase the frequency and watch the display of the microphone signal on the oscilloscope. The microphone signal is a measure of the pressure amplitude of the sound wave. At the resonant frequency, the microphone output goes to a maximum. The first one is the fundamental (n = 1). Continue to increase the frequency of the oscillator and locate the resonant frequencies up to n = 10. Record your data in a table. 3. Read and record the length, L, of the tube. 4. Read and record the room temperature from a thermometer in the lab. Data Analysis 1. From equation (2), a graph of fn against n should be a straight line. Plot the graph and find the slope. 2. Calculate the velocity of sound v from the slope of the graph. 3. For comparison, calculate v from equation (5). 4. Calculate the percent difference between your experimental value of v and the (accepted) value obtained in step 3. If the experiment was done carefully, this difference should be less that 5%. 5. State the sources of error that would contribute to this difference. 6. A method to find γ of a gas is to measure the velocity of sound in that gas. Use your measured velocity of sound in air as found in step 2 and calculate the value of γ for air using equation (3). Use the equation below to find the density of air in the lab: ρ (g/cm3 ) = 0.001293 H (cmHg) 1 + 0.00367 t (°C) 76 Compare your value of γ with the accepted value of 1.403 for air. © KFUPM – PHYSICS revised 29/11/2007 14 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 (OPTIONAL) Part B. Variation of sound wave pressure along a standing wave in a tube closed at both ends. Procedure Here we shall measure the pressure amplitude as a function of distance x from the sound source when the fundamental standing wave pattern is set up in the tube closed at both ends. 1. Set the frequency of the oscillator at the resonant frequency of the first harmonic, as found in Part A. Do not change this frequency setting for the rest of this experiment. 2. Move the stainless steel rod until the microphone is 1 cm from the sound source (x = 1 cm) and measure the peak-to-peak voltage (PPV) of the microphone signal on the oscilloscope. Measure the PPV as a function of x as you move the microphone to the other end until it is level with the end plug. It is recommended that x should be changed in steps of 2 cm. Make a table and record the PPV and the x values. Data Analysis Note that the microphone actually measures the sound intensity and not the amplitude of the wave. The amplitude is proportional to the square root of the intensity. 1. Plot a graph of √ PPV against x. 2. What you have plotted is the shape of the upper half of the fundamental standing wave in the tube shown in Figure 2 for n = 1. To get the complete shape plot the same √ PPV values with the sign changed against the x values on the same graph. 3. From your graph find the distance between the pressure node and the nearest antinode. 4. From the result of step 3, calculate the wavelength of the standing wave and then the velocity of sound. Exercise Draw diagrams similar to those shown in figure 2 for the tube open at one end. Find the expression for the frequencies in this case. © KFUPM – PHYSICS revised 29/11/2007 15 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 SPECIFIC HEAT OF A METAL Purpose To determine the specific heat of aluminium. Background The specific heat of a substance is the quantity of heat necessary to raise a unit mass of the substance by a unit temperature difference. The specific heat of solids varies with temperature in a characteristic way as indicated in Figure (1). It is seen that for T >> θ D (the so-called Debye temperature) the specific heat tends to a constant value per mole which is the same for all solids). However for T<< θ D the specific heat becomes a strong function of temperature. In this experiment we will measure the amount of liquid nitrogen (LN2) boiled off when a mass of aluminium is cooled to LN2 temperature. The basic equation is, assuming no net gains or losses of heat, heat lost = heat gained mc∆T = ∆m N LN (1) where © KFUPM – PHYSICS revised 29/11/2007 16 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals m c ∆T ∆ mN LN = = = = = = = PHYS102 mass of aluminium (shot + bucket) unknown specific heat of aluminium room temperature (in Kelvin) - LN2 temperature room temperature (in Kelvin) - 77.35 K. mass of LN2 boiled off latent heat of LN2 47.7 cal/g Putting into equation (1) measured values of the various quantities we can find the specific heat. Table 1 ________________________________________________________________________ Material θ D (Kelvin) Classical Condition Expected Behaviour Specific Heat ________________________________________________________________________ Aluminium 428 0.215 cal/g Co T<< θ D Non-classical _______________________________________________________________________ From Table 1 one can see that the temperature of LN2 (T = 77.35 K) is much less than the Debye temperature of aluminium. Thus we would expect to measure a value of the specific heat which is less than the classical value. Data In this experiment we will be using liquid nitrogen produced in KFUPM's Cryogenic Facility. WARNING LIQUID NITROGEN IS AN EXTREMELY COLD SUBSTANCE. HANDLE IT WITH CARE. PARTICULARLY AVOID TOUCHING METAL SURFACES WHICH ARE COOLED TO ITS TEMPERATURE. 1. Record the room temperature using the thermometer hanging on the wall. 2. Measure the mass of the bucket with aluminium shot. 3. Place the dewar containing liquid nitrogen on the balance. Notice that the liquid nitrogen is boiling away sufficiently fast to prevent you from obtaining a stable value for its mass. However, we will be able to get around this difficulty by measuring the values of mass vs. time. 4. Start the stopwatch. Choose a value of the mass a few grams lighter than the present value of the mass of dewar and set it into the balance. In a short time as the LN2 boils away the dewar will come to balance. Record this time. Set in a new lower value of mass. Continue recording the times until you have, say, 5 values. (NOTE: KEEP YOUR STOPWATCH GOING). © KFUPM – PHYSICS revised 29/11/2007 17 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 5. Slowly immerse the bucket of shot into the LN2 so that LN2 does not spill, HOLDING THE BUCKET BY THE NYLON STRINGS. After the boiling ceases, repeat the measurements in step (4). 6. Remove the bucket. DO NOT TOUCH THE BUCKET OR THE SHOT. **DON'T STOP THE STOPWATCH DURING THE MEASUREMENTS (STEPS 4 -5)** Calculations 1. Subtract m (mass of aluminum shot + bucket) from your mass values taken after the bucket was immersed. Plot the values of (dewar + LN2) mass so obtained vs time. Your graph will be similar to the one below. 2. Use equation (1) to obtain the specific heat of aluminium. Discussion and Error Analysis 1. Is your value for Cal significantly lower than the classical value given in Table 1, as expected? 2. If we average the temperature-dependent specific heat of aluminium between LN2 and room temperature, theory predicts that CAl = 0.153 cal/g Co How does your value agree with this? What is the percent difference? 3. Given that the Debye temperature of lead is 105 K, what kind of behaviour (nonclassical or approximately classical) would you expect for lead? Explain why. © KFUPM – PHYSICS revised 29/11/2007 18 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 PERFECT GAS LAW Purpose To study the perfect gas law. To measure the change of pressure of a fixed volume of air when its temperature is changed. To determine the absolute zero of temperature and the gas constant. Background The pressure of a fixed volume of all dilute gases is found to vary linearly with temperature according to Pt = Po (1 + A t) (1) Here Pt is the pressure at temperature t, and Po is the pressure at the zero (t = 0) of the particular scale used (Fahrenheit, Celsius, etc.). A is some constant. Look at equation (1) and notice that for t = - 1/A, Pt = 0. At temperatures lower than this point the pressure would become negative, which is impossible. Thus this temperature is called the absolute zero. Since equation (1) is the same for all gases, it can be used to define a new scale of temperature, the absolute or Kelvin scale. Call temperatures on this new scale T and set t = - 1/A + T (2) Then equation (1) becomes P = Po A T (3) If the volume of the gas is now varied it is found that the complete equation may be written (for n moles of the gas) P= nR T V (4) This is the perfect gas law, where R is the universal gas constant. Method In this experiment we will measure the pressure of a fixed volume of air, V, in a bulb, B, using a pressure gauge, G. The pressure will be measured as a function of temperature t. © KFUPM – PHYSICS revised 29/11/2007 19 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Gauge G Valve H Bulb B Volume V Volume V Figure 1 Initially V is at pressure P1, and room temperature t1. (NOTE THAT THE PRESSURE MEASURED FROM THE GAUGE G SHOULD BE in kPa = 1*103 Pa = 1*103 N/m2). The bulb B is then immersed in a mixture of ice and water. At this temperature t2 the pressure will decrease to P2. The bulb is then immersed in a hot water bath at specific temperatures t3, t4..., and the corresponding pressures P3, P4…, are read from the gauge G. Procedure You should draw up a suitable table in which to record the data. 1. Read off the room temperature t1, in degrees Celsius, from the thermometer provided and the pressure P1 as read from the pressure gauge G (in kPa). 2. Put some ice into the stainless steel beaker provided and fill it one-third with water. Hold the apparatus by the handle H and immerse the bulb B in the ice-water mixture. Allow the temperature of the bulb to stabilize and wait until the needle on the pressure gauge stops moving. Record the temperature t2 of the mixture using the thermometer provided and pressure P2 from the pressure gauge, G. . © KFUPM – PHYSICS revised 29/11/2007 20 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 3. Remove the bulb from the ice water and let it warm up to room temperature. Check that the pressure gauge reading returns to pressure, P1. 4. Fill the stainless steel beaker about one-third full of water. Place the beaker on the hot plate and switch it on. 5. Use the thermometer to monitor the temperature of the hot water. When the temperature reaches about 45 oC, CAREFULLY immerse the bulb in the beaker. Take care not to touch the wall of the beaker with the bulb. Wait a few minutes and then record the pressure P3 and the temperature t3. 6. Remove the bulb from the stainless steel beaker. When the water starts boiling, CAREFULLY immerse the bulb into the boiling water, and when the pressure has stabilized, read and record the temperature t4 and pressure P4. Then remove the bulb from the stainless steel beaker and switch off the hot plate. Data Analysis 1. Restricting the range of the temperature axis from 0°C to 100°C, plot Pt (in Pa) vs. t (in °C) and draw the best straight line through your data points. 2. From equation (1), the slope and y-intercept of the graph of Pt vs. t (in °C) are Po A and Po, respectively. Obtain the slope and the y-intercept from the graph and hence calculate value of A. The absolute zero is then equal to -1/A in the Celsius system. This is the temperature at which Pt is zero. 2. According to equation (4) the slope of Pt vs. t equals n R/V. Find the experimental value of R. We will not measure n/V but instead take it to have the room temperature value n/V = 40.89 mole/m3. 3. Calculate the percentage difference between your value of the absolute Zero and the accepted value of –273.15 °C, and between your value of R and the accepted value of 8.31 J/mole. K. 4. List and discuss the major sources of error. © KFUPM – PHYSICS revised 29/11/2007 21 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 OHM'S LAW-1 Purpose To study Ohm's law as applied to a "linear" DC circuit. Theory The resistance R of a conductor depends upon a number of factors including its nature, dimensions and even temperature. Most conductors have a constant resistance at constant temperature, so that the current I produced through the conductor is directly proportional to the voltage V across it. The relationship called Ohm's law states that I = V / R (R constant). (1) The resistance R of the conductor is measured in Ohm ( Ω ), thus 1 Ω = 1 Volt / 1 Ampere. The equivalent resistance Rs of two resistances R1 and R2 connected in series, as shown in Figure 1-a, is given by Rs = R1 + R2. (2) In the parallel configuration, shown in Figure 1-b, the equivalent resistance Rp is given by 1/Rp = 1/R1 + 1/R2, or Rp = (R1 R2) / (R1 + R2). (3) For some conductors the current I is not proportional to V. For example, for metallic conductors, if I is very large it will cause heating of the conductor so that it will be no longer proportional to V. In that case, the conductor is called a non-ohmic one since it no longer obeys Ohm's law. © KFUPM – PHYSICS revised 29/11/2007 22 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Experimental Set-up The basic circuit used in this experiment comprises a d.c. power supply, an ammeter, a voltmeter and the resistance(s) to be studied, as shown in Figure 2. The ammeter (here a multimeter is used as an ammeter) will measure the current I flowing through resistor R. It is therefore connected in series with the resistor. The voltmeter, which will measure the potential difference V across the resistor, is connected in parallel with the resistor. The polarities of the power supply, the ammeter and the voltmeter are clearly indicated in Figure 2. They must be respected or one would run the risk of damaging either the ammeter or the voltmeter, or both. Procedure A. Measurement of Single Resistance 1. Connect the circuit as shown in Figure 2. R1 represents one of the two unknown resistors provided. 2. Switch on the power supply and vary the input voltage in regular steps between 0 and 5 volts. Record the voltage across R1 and the current I. Note: Do not use the dial on the power supply to read the voltage, use the voltmeter. 3. Plot a graph of I versus V and verify that a straight line is obtained. Find the value of R1 from the slope of the graph. 4. Replace R1 by R2. Measure the current I for, say, three values of the voltage V. Calculate the average value of R2. © KFUPM – PHYSICS revised 29/11/2007 23 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals B. PHYS102 Measurement of Resistances in Series and Parallel 5. Connect R1 and R2 in series (see Figure 1-a) and, using the circuit shown in Figure 2, measure the current I for, say, three values of the voltage V. Calculate the average value of the equivalent resistance Rs. 6. Connect R1 and R2 in parallel (see Figure 1-b) and, using the circuit shown in Figure 2, measure the current I for, say, three values of the voltage V. Calculate the average value of the equivalent resistance Rp. 7. Using the values of R1 and R2 obtained in part A, calculate the equivalent series and parallel resistances, Rs and Rp using equations 2 & 3. Compare these results with those found in steps 5 and 6, respectively. 8. Calculate the % difference between the calculated values and the experimental values. Exercise Assume that the percentage error in each of R1 and R2 is 5%, ∆R1 / R1 (100) = ∆R2 / R2 (100) = 5% , use your results for R1 and R2 to calculate: i.e. a. The maximum possible error in Rs, i.e., ∆Rs when the two resistances are connected in series. b. The maximum possible error in Rp, i.e. ∆ Rp when the two resistances are connected in parallel. (Note that when calculating ∆Rp from Rp = R1R2/(R1+R2), you have to use: ∆R p = ∂R p ∂R1 ∆R 1 + ∂R p ∂R 2 ∆R 2 . You can’t use the shortcut: Z=A/B and relationship ∆Z/Z=∆A/A+ ∆B/B, can you explain why? ). © KFUPM – PHYSICS revised 29/11/2007 24 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 OHM'S LAW-2 Purpose To study Ohm's law as applied to a "linear" DC circuit. To show the behavior of some "nonlinear" circuit elements which do not obey Ohm's law. Theory The resistance R of a conductor depends upon a number of factors including its nature, dimensions and temperature. Most conductors have a constant resistance at constant temperature, so that the current I produced through the conductor is directly proportional to the voltage V across it. The relationship called Ohm's law states that I = V / R .......................................... (1) where R is a constant of proportionality. The resistance R of the conductor is measured in Ohms, ( Ω ); thus 1 Ω = 1 Volt / 1 Ampere. For some conductors the current I is not directly proportional to V. For example, for metallic conductors, if I is very large it will cause heating of the conductor and I will no longer be proportional to V. In that case the conductor is called non-ohmic since it no longer obeys Ohm's law. Heating of the metallic conductor results in the increase of its resistance. The resistance of a conductor varies in an approximately linear fashion with temperature over a limited temperature range according to the expression R = R0 [1 + α (T − To )] ..........................(2) where R is the resistance of the conductor at some temperature T, Ro is the resistance at some reference temperature To, say room temperature, and α is called the temperature coefficient of resistance. Note that α is strictly not constant but may vary slightly with temperature. Using equation (2) and solving for ∆ T = (T-To ), we get ∆ T = T-To = (R-Ro ) / (αRo ) ..................(3) Experimental Set-up The basic circuit used in this experiment comprises a d.c. power supply, an ammeter, a voltmeter and the resistor to be studied, as shown in Figure 1. © KFUPM – PHYSICS revised 29/11/2007 25 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 The ammeter (here a multimeter is used as an ammeter) will measure the current I flowing through resistor R. It is therefore connected in series with the resistor. The voltmeter, which will measure the potential difference V across the resistor, is connected in parallel with the resistor. The polarities of the power supply, the ammeter and the voltmeter are clearly indicated in Figure 1. They must be respected or one would run the risk of damaging either the ammeter or the voltmeter, or both. Procedure A. Measurement of Resistance 1. Connect the circuit as shown in Figure 1. R represents the unknown resistor. 2. Switch on the power supply and vary the input voltage in regular steps. Record the voltage drop V and the current I. Note: Do not use the dial on the power supply to read the voltage; use the voltmeter. 3. Plot a graph of I versus V and verify that a straight line is obtained. Find the value of R1 from the graph. B. Voltage-Current Relationship for a Flashlight Bulb 1. Using the circuit of Figure 1, replace R by the flashlight bulb. 2. Use this set-up to investigate how the resistance of the lamp varies with the voltage drop V across the lamp. Record the current I for the values of V given in the table: © KFUPM – PHYSICS revised 29/11/2007 26 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals Voltage Drop V (volts) PHYS102 Current I(A) R= V/I Ω 0.1 0.2 0.4 0.6 0.8 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 3. Plot a graph of resistance R against voltage drop V for the lamp. 4. From your graph find the resistance R of the lamp when it is operating at 12 volts. 5. Calculate the power P of the lamp when operating at 12 V (P = V2/R). 6. From your graph find the resistance Ro of the lamp at room temperature (i.e. when the voltage drop across it is zero). 7. Find the room temperature To from the thermometer hanging on the wall of this lab. 8. Using equation (3) and the above data (steps 5 to 7), find the value of T of the lamp when it is operating at 12 volts (the value of the temperature coefficient for pure tungsten, as given by Serway, is 4.5 x 10-3K-1). 9. The accepted value of the filament temperature at the maximum rated power is about 3000 K. Explain the difference between your value and the accepted value. 10. The power rating for this tungsten filament is given as "12 V, 24 W". Explain why this is the maximum operating power for this lamp, (the melting point of the tungsten is 3695 K). © KFUPM – PHYSICS revised 29/11/2007 27 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 WHEATSTONE BRIDGE Purpose 1. To introduce bridge circuits and null detection method to measure the resistance of a conductor. 2. To measure the resistivity of several conductors. 3. To determine the variation of the resistance of a conductor with its length and crosssectional area. Theory A. The Wheatstone Bridge Accurate measurements of an unknown resistance can be performed by comparing it with standard resistances (resistances which have been previously determined to sufficient accuracy) in some kind of bridge circuit, such as the Wheatstone bridge. The conventional diagram of a Wheatstone bridge is illustrated in Figure 1. The combination is called a "bridge" because the voltmeter is bridged between two parallel branches, DAC and DBC. The bridge is said to be "balanced" when the resistances are so adjusted that points A and B are at the same potential. This is indicated by Digital Multi Meter (DMM) reading zero. (In practice, the voltage reading will swing between + and – values). © KFUPM – PHYSICS revised 29/11/2007 28 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 When the bridge is balanced, the current I1 through resistances R1 and R2 is the same, and current I2 is the same in resistances R3 and R4. Since the points A and B are at the same potential, the voltage across R1 equals the voltage across R3. Thus V1 = V3 and, therefore, I1 R1 = I2 R3 (1) Similarly, the voltage across R2 equals the voltage across R4; hence V2 = V4 and, therefore, I1 R2 = I2 R4 (2) Dividing these equations, we find that the currents cancel so that R1 / R2 = R3 / R4 (3) which is the balance condition. Equation (3) could be written as R1 = R2 R3 / R4 (4) Hence an unknown resistance may be measured by comparison with three known resistances. B. Resistance and Resistivity The resistance of a wire of length L and cross-sectional area A may be calculated from R = ρL / A (5) where R is the resistance of the wire and ρ is the "resistivity" (or specific resistance) of the material. The resistivity depends on the material in question and on its temperature. It can be obtained by writing equation (5) as ρ = RA / L (6) In the metric system the resistivity of a material is numerically equal to the resistance of a piece of material one meter in length and one square meter in cross-sectional area. Experimental Set-up The experimental set-up is shown in Figure 1. In this circuit the unknown resistance is R1. R2 is a known resistance, R3 and R4 are resistance boxes. © KFUPM – PHYSICS revised 29/11/2007 29 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Procedure A. Measurement of a Resistance 1. Connect the resistors provided in the circuit (see Figure 1). 2. Obtain a rough idea of the resistance R1 by making R4 = R2. 3. Equation (4) indicates that balance is achieved when R1 = R3. Write down the value of R3. This is the approximate value of R1. 4. The value you get from the previous step is not very accurate, but you can obtain a better estimate of R1 by making R4 = 10 R2. Again, vary R3 until balance is obtained and then use the equation 4 to find the new value of R1. 5. Repeat step 4 by making R4 = 100 R2. This gives the best value of R1. 6. Question: Can the accuracy of this bridge circuit be increased without limit? Make R4 = 1000 R2 and try to find a balance. Record the value of R1. B. Measurement of Resistivity You are provided with a set of five spools. The wire in each spool is made of a certain material, and has specific diameter and length (see Table 1 for details). 7. Use the Digital Multi Meter (DMM) to measure and record the resistances of the spools 1 to 5. 8. Using equation (6) and the data provided in table 1, calculate the resistivity of each spool. 9. Draw up a table that enables you to answer the following questions: (a) Considering the two spools of the same material and diameter, how does the resistance vary with the length? Write the relationship. (b) Considering the two spools of the same material and length, how does the resistance vary with the radius? Explain. (c) Why is the resistivity of spool 1 and 5 different? Explain. (d) Calculate an average value for the resistivity of copper (material of spools 14). (e) The accepted value for the resistivity of copper is 1.7 x 10-8 Ω .m. Calculate the percent difference. © KFUPM – PHYSICS revised 29/11/2007 30 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals (f) PHYS102 Discuss the major sources of error in this experiment? Table 1: Characteristics of the spools © KFUPM – PHYSICS revised 29/11/2007 Spool Number Material Length (m) Diameter (cm) 1. 2. 3. 4. 5. Cu Cu Cu Cu Ni-Ag 10 10 20 20 10 0.0644 0.0322 0.0644 0.0322 0.0644 31 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 FROM GALVANOMETER TO VOLTMETER Purpose The purpose of this experiment is to apply Ohm's law in the determination of a galvanometer resistance and to study how a moving coil galvanometer circuit can be modified to construct a voltmeter. Background All moving-coil meters are based on the d'Arsonval galvanometer. The basic components of this galvanometer are shown schematically in Figure (1). A micro-ammeter is just a galvanometer calibrated to read in micro-amps. In this experiment you will be using a 500 µ A galvanometer to construct a voltmeter. This galvanometer gives a full-scale deflection for a current Ig = 500 µ A. In another experiment you will learn how to construct an ammeter. 1 µA = 1 micro-amp = 10-6 amp. 1mA 1 milli-amp = 10-3 amp. = A galvanometer can be converted to a voltmeter by connecting a high resistance Rh in series with the galvanometer as shown in Figure (2). When the voltmeter is connected across a potential difference (V), the same current (Ig) flows through both Rh and Rg, so that V = ( Rh + Rg ) Ig © KFUPM – PHYSICS revised 29/11/2007 32 (1) Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 If (Ig) is the current required to give full-scale deflection, then (V) can be made into any required value by adjusting the value of the high resistance (Rh). The trade name Avometer stands for (Amp-Volt-Ohm-Meter). The instrument is built around a sensitive galvanometer. The quantity to be measured and the range (full scale reading) can be selected with a circular dial. For the voltmeter function the dial switches appropriate high resistances in series with the galvanometer. The instrument can also be used to measure resistances, currents and alternating currents and voltages, but these will not be required for this experiment. ___________________________________________________________________________ Caution: The galvanometer used in this experiment is sensitive. It can be damaged by a momentary overload of current. As a precaution have all circuits checked by your instructor before you connect it to the voltage. ___________________________________________________________________________ Part 1. Measurement of Galvanometer Resistance, Rg In this Part 1 we measure (Rg) for the 500 µ A galvanometer. We use an indirect method which does not damage the galvanometer by passing too much current through it. Note: Do not try to measure the resistance of any galvanometer with an avometer. 1. Set one decade box R1 at 90,000 ohms and connect it in series with the 500 µ A galvanometer and the voltage source as shown in Figure (3.a). © KFUPM – PHYSICS revised 29/11/2007 33 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 2. Adjust R1 until the galvanometer gives a full-scale deflection. 3. Keeping R1 fixed, connect the second decade box R2 in parallel with the galvanometer as shown in Figure (3.b). 4. Adjust R2 until the galvanometer reads half of the full scale. This means that half the original current now flows through R2, so that the resistance of the galvanometer is © KFUPM – PHYSICS revised 29/11/2007 34 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 approximately equal to R2. (This approximation assumes that R1 is much greater than R2 or Rg). Record the R2 value in your report. Part 2. To Convert a 500 µ a Galvanometer into a Voltmeter In this Part 2 the 500 µ A galvanometer will be converted into a 10 volt range voltmeter. 1. Calculate the high resistance (Rh) that must be connected in series with the 500 µ A galvanometer to give 10 volts potential drop across both when the current is 500 µ A. (See Figure 2 and equation (1)). 2. Use one decade box for this resistance (Rh) and check the calibration of your constructed voltmeter by means of the circuit shown in figure (4). In this circuit (V) is the avometer (assumed to be correct) and R2 is the second decade box, set initially at 9999 ohms. (This resistance box acts as a protective device and also as a fine adjustment for controlling the voltage). 3. Adjust the power supply to give readings of 1,2,3,.., 9 volts on the constructed voltmeter, i.e., on the galvanometer G. For each value, read and record the correct voltage on the avometer, V. 4. Plot a calibration graph, i.e., plot the volts values from G versus the avometer readings. Now, your constructed voltmeter combined with the calibration graph can be used as an accurate voltmeter. 5. Write a brief conclusion for this Part. © KFUPM – PHYSICS revised 29/11/2007 35 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Exercises 1. In Part 1, what approximation has been made in assuming that Rg = R2. Make a rough estimate of percent error introduced by this approximation. 2. Calculate the resistance of your homemade voltmeter. 3. From your results of Part 2 explain and discuss types and sources of errors in the experiment. © KFUPM – PHYSICS revised 29/11/2007 36 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 SLIDE-WIRE POTENTIOMETER Purpose 1. To establish the linear voltage/displacement relationship of the potentiometer. 2. To measure the emf of an unknown cell using the slide-wire potentiometer. Introduction The potentiometer is a device that is used to measure an unknown emf, Ex, by using a known emf. Figure 1 shows the essential components of a slide-wire potentiometer. Point d represents a sliding contact used to vary the potential difference between points a and d. The other required components are a galvanometer, a power supply with emf E0, a standard reference cell with an emf, Es. Ex or Es +_ c b G Ix I Rx or Rs I - Ix d I + a Eo Figure 1. Applying Kirchoff's rule to loop abcd gives Ex +(I - Ix) Rx = 0. The sliding contact d is now adjusted until the galvanometer reads zero. Under this condition, the current Ix in the galvanometer and in the unknown cell is zero. Then Ex = I Rx (1) Next the cell of unknown emf is replaced by a standard cell of known emf, Es, and the above procedure is repeated. That is, the moving contact d is varied until a balance is obtained, then Es = I Rs (2) where Rs is the resistance between points a and d, when balance is achieved. From equations (1) and (2) one gets R Ex = x Es Rs © KFUPM – PHYSICS revised 29/11/2007 (3) 37 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 If the resistivity of the wire is ρ and its cross sectional area is A, Rx and Rs in terms of Lx and Ls, (the lengths of slide-wire where balance was achieved, when the unknown and the standard L L cells were used) are R x = ρ x and Rs = ρ s . Then equation (3) becomes A A L Ex = x Es Ls (4) Part 1. Establishing linear voltage - distance relationship for potentiometer - Power Supply + L V Figure 2. 1. Set up the potentiometer circuit shown in figure (2) and set V on the power supply to approximately 2 volts. Do not use very large V value. 2. Use the digital voltmeter to measure V as a function of L for at least 10 points evenly spaced along the wire. 3. Plot a graph of V versus L. 4. Does your graph of V versus L indicate linear relationship? Discuss. Part 2. Measurement of emf of a cell - Power Supply + Lx or Ls + - 15 kΩ G Ex or Es Switch © KFUPM – PHYSICS revised 29/11/2007 Figure 3. 38 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals 1 PHYS102 Set up the circuit shown in Figure 3. Note that the 15 KΩ resistance is in the circuit to protect the standard cell from having large currents drawn from it when the tapping point B is far from balance, and also to protect the galvanometer from excessive current which could damage the pointer and the suspension. 2. Set V on the power supply to approximately 2 volts (measure this voltage while the circuit is connected) and adjust the position of the tapping point on the slide wire, seeking null balance (zero current through the galvanometer). 3. When you have a null balance, increase the null sensitivity by closing the switch S and then retire the adjustment at the tapping point to achieve a more sensitive null balance. Record the length Ls. 4. Replace the standard cell with the cell whose emf is to be measured (carbon-zinc cell) and find its null balance point. Call the length of the wire Lx and determine the emf Ex for the carbon-zinc cell using equation (4). Es = 1.018V. 5. Using a digital multimeter determine the potential across the terminals of the cell. Explain the difference between this value and the value you found in step (4). © KFUPM – PHYSICS revised 29/11/2007 39 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 CAPACITORS IN SERIES AND PARALLEL Purpose 1. 2. To measure the capacitance of a capacitor. To investigate the capacitance of capacitors in series and in parallel. Introduction The performance of many circuits can be predicted by systematically combining various circuit elements in series or parallel into their equivalents. For capacitors the equivalent capacitance for series and parallel combinations is as follows: 1/Cs = 1/C1+1/C2 + ....+ 1/Cn C1 ≡ C2 Cs For two capacitors 1/Cs = 1/C1 + 1/C2 Cn or Cs = C1C2/(C1 + C2). (1) SERIES ---------------------------------------------------------------------------------------------------------------------- Cp = C1 + C2 + .... + Cn C1 C2 Cn ≡ Cp For two capacitors Cp = C1 + C2 . PARALLEL © KFUPM – PHYSICS revised 29/11/2007 40 Department of Physics Dhahran 31261 (2) King Fahd University of Petroleum & Minerals PHYS102 Method If a capacitor is charged to a certain voltage, and then disconnected from the voltage source, the voltage on the capacitor will stay at the same value for a long time (determined by the leakage resistance of the capacitor). Vo Vo Vo Vo ….retains voltage Vo Capacitor charged to Vo C R C C 0 ….discharges However, if the capacitance is connected to a resistance R, it will discharge; the time it takes to discharge is governed by R and C. Circuit theory indicates that the voltage at time t after the voltage source is disconnected is: V = Vo e-(t/RC) (3) Here Vo is the initial voltage and e is the base of natural or Naperian logarithms, e = 2.71828... In this experiment we will measure the "1/k" time, T1/k, that is, the time needed for the voltage to change from Vo to Vo/k. Here k is a number in the range 1 < k < ∞ . You are perhaps familiar with certain T1/k values: "half-life" T1/2 = 0.693 RC (k = 2) (4) "1/e time" T1/e = RC (k = e) In this experiment your instructor will assign each student a different value of k. Data 1. Chose one of the capacitors given and connect the circuit shown below. © KFUPM – PHYSICS revised 29/11/2007 41 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Note: IN THIS EXPERIMENT WE WILL BE USING POLARIZED CAPACITORS, THAT IS, CAPACITORS WITH A POSITIVE AND NEGATIVE TERMINAL. IT IS VERY IMPORTANT THAT CARE BE TAKEN TO HOOK THESE UP AS IN THE DIAGRAMS, OTHERWISE THEY MAY BE DAMAGED. IF YOU ARE UNSURE HOW TO DO THIS CONSULT THE INSTRUCTOR. switch S 47Ω a voltmeter Power supply + _ C + _ 15.0 x 103 Ω 15 V range b 2. With the switch closed, turn on the power supply and adjust it until the voltmeter reads some convenient voltage, say 10 volts. This is Vo . 3. Open the switch and start the stopwatch. Measure the time for the voltage to fall from Vo to Vo/k. This is T1/k. 4. Use equation (3) to derive a relation between T1/k and RC, for your assigned value of k, similar to equations (4). Use the derived equation and the value R = 15.0 x 103 ohms to calculate the capacitance of the first capacitor. (R is the effective resistance of the voltmeter.) a a 5. Repeat with the second capacitor. + _ 6. Repeat with both capacitors in series. 7. Repeat with both capacitors in parallel. + _ + _ + b b SERIES PARALLEL Optional V vs t for a Discharging RC Circuit Connect the resistor that you are provided in series with the voltmeter. In this case discharging takes place much more slowly. Collect data to make a curve of V vs t in the following way. Charge the capacitor to the full range of the voltmeter. Open the switch and start your stopwatch. Record the reading of the voltage across the capacitor when t = 3 s. © KFUPM – PHYSICS revised 29/11/2007 42 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Charge the capacitor again to the full range of the voltmeter. Open the switch and start your stop watch. Record the reading of the voltmeter at t = 6 s. Record this process for t = 9 s, t = 12 s and higher values of t. Plot curves of V vs. t and ln V vs. t. Draw a smooth curve through your point. The curve of lnV vs. t should be a straight line. This can be proved by taking the logarithm of both sides of eq. 3. ln V(t) = ln Vo – ( t / RC ) Find the RC time constant from your graph. This is the time where V has fallen to 1/e of its initial value. It is also equal to the negative inverse of the slope of the line in lnV vs. t. Discussion and error analysis 1. Calculate the expected series capacitance based on equation (1) and your measurements in parts 4 and 5; what is the percent difference between this and the measured value in part 6? 2. Repeat the calculations in (1) for the parallel combination. 3. What are the major sources of error in this experiment? 4. Give a mathematical or physical argument why k cannot have values in the range 0 ≤ k ≤ 1. . 5. Derive an expression for the current through the voltmeter while the capacitor is discharging. 6. What will be the final current through the voltmeter. © KFUPM – PHYSICS revised 29/11/2007 43 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 INTRODUCTION TO THE CRO Purpose To learn how to operate a Cathode Ray Oscilloscope (CRO) and to use it in studying simple circuits. Introduction The CRO is one of the most useful electronic measuring devices. It is essentially a glorified voltmeter and a chart-recorder whose output is displayed as a function of time on a television-like screen rather than on a meter. The heart of a CRO is the cathode ray tube whose essential components are shown schematically in figure (1). This is a tube which has an 'electron gun' at one end, and a large fluorescent screen at the other. It is not necessary to have an intimate knowledge of electron optics in order to appreciate the operation of the CRT; suffice it to say that a set of accelerating and focusing electrodes immediately in front of the filament of the electron 'gun' draw electrons from the heated filament and focus the stream of electrons into a well-defined beam. When this beam strikes the fluorescent screen (also referred to as the scope face), a 'spot' appears. Before the beam reaches the screen, it passes between two sets of deflecting plates. When voltages are applied to these plates, the beam is deflected (either up or down by the Y plates, or left and right by the X plates). A negative voltage will cause a deflection in one direction, and a positive voltage a deflection in the other. © KFUPM – PHYSICS revised 29/11/2007 44 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 The voltage to be measured is generally applied to the Y plates, while an internal 'sweep' generator is connected to the X plates. This causes the spot to move horizontally in a linear fashion so that the 'X axis' now represents time, while the Y direction represents the amplitude of the signal. (This sweep generator is discussed in detail below). The deflection plates The electrons forming the electron beam are accelerated by the potential difference V between the cathode and the second anode. If the charge on each electron is -e, the electrons will have a final velocity (v) given by 1 2 mv = e ∆V 2 (1) where m is the mass of the electron. When a voltage is applied to the deflection plates, an electric field (E) is created between them whose magnitude is given by E=V d (2) where (d) is the separation between the plates. An electron moving through this electric field is accelerated in a direction opposite to the electric field experiencing a force F = |e| E = |e| V d As a result, the beam is deflected from its original path. Also, the deflection at the screen is proportional to the deflecting voltage. For the Y-plates Y α Vy. The saw-tooth generator If a voltage which increases linearly with time is applied to the X plates, the spot will move across the screen linearly with time, painting out a 'time' axis. The saw-tooth generator generates a waveform such as that shown in figure (2). The maximum corresponds to a maximum deflection to the right of the screen, and a minimum to maximum deflection to the left of the screen. When the maximum voltage is reached, the voltage drops to a minimum and the voltage starts increasing all over again. At low sweep speeds, the spot can be seen to travel across the face of the CRO, while for high sweep speed only a line is visible. The time base control knob is marked in units of time/scale division, e.g. 100 m sec/cm: Hence it takes the spot 100 m sec to travel 1cm across the screen. (Hence the term time base). The rate at which the spot travels across the screen can be changed at will by changing the setting of the TIME BASE control knob. © KFUPM – PHYSICS revised 29/11/2007 45 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 The dual-beam oscilloscope Fig. 1 shows the CRT for an 'ordinary' oscilloscope. A Dual-beam oscilloscope such as the one used in this experiment (see Fig. 3) has two electron guns with separate sets of deflecting plates so that two different signals can be displayed simultaneously on the screen. This in no way complicates the operation of the scope, but vastly expands its usefulness. Each of the 'Y inputs' is connected to its own amplifier (i.e. channel) so that the two signals can undergo different amplifications. It is important to note, however, that both traces are driven by the same time base generator, so that they both travel at the same speed. (Some dual-beam scopes have separate time bases for each channel). Figure (3) shows the CRO controls which are most frequently found on a commercial oscilloscope. Your instructor will demonstrate the function of these controls at the beginning of the laboratory period. Study (1): Familiarization with CRO controls and frequency measurements 1. Switch on the CRO. Find the two traces: Pull out the trig-level button and set at maximum (anti-clockwise) so that the traces are 'free running'. If a trace does not appear, use the ↔ and ↑ knobs to find the trace. 2. Set CRO time base to 200 µs/cm and the vertical sensitivity for channel 1 to 5V/cm. © KFUPM – PHYSICS revised 29/11/2007 46 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 3. Set the trig. Switch to channel 1 and make sure that channel 1 and the TV-Ac switch are both on AC. Also set (+-) switch to (+) and push in the trig. level button. 4. Set the signal generator to 1 kHz and the sine wave amplitude control to maximum. 5. Connect the output sine waveform of the signal generator to channel 1 of the CRO. © KFUPM – PHYSICS revised 29/11/2007 47 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 6. Adjust intensity and focus controls on CRO (note that the trace should never be too bright, as a bright trace could eventually damage the phosphor on the screen). Move the position knob until you can see the start of the trace. 7. The CRO is now giving a voltage-versus-time plot of the output of the signal generator (see figure 4). 8. Measure the number of centimeter scale division on the CRO screen occupied by one cycle. If one cycle occupies 5 divisions with the time base set at 200 Time Duration = Horizontal distance (5cm) X 200 = 103 µs Frequency = = 10-3 s/cm, therefore s/cm seconds 1 1 = − 3 = 1 kHz Time Duration 10 This frequency is that of the signal generator. 9. Vary the frequency setting of the signal generator and the time scale on the CRO and roughly check the accuracy of the time axis. Study (2): The signal generator as a voltage generator From the previous study the voltage-versus-time display on the CRO represents an alternating voltage versus time. This alternating voltage alternates from (+A) to (-A) in times which are short compared to one second (see figure 4). The peak-to-peak voltage (p-p voltage) is the voltage difference from the top to the bottom of the sine wave. When using an AC voltmeter, it would read the voltage value represented by the dotted line on figure (4). © KFUPM – PHYSICS revised 29/11/2007 48 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 This voltage is A / 2 . Thus the effective voltage of an alternating signal whose voltage peaks are +A and -A is given by A / 2 . This is also called the root mean square (rms) voltage. Consider the circuit shown in figure (5), using Kirchhoff's second Law. εo = I(Ro + R) (1) (This assumes that negligible current enters the CRO) Also V = IR (2) Eliminating I from (1) and (2) or εo = (Ro + R) V R (3) R = Ro + R V εo (4) In this study we will measure the peak-to-peak voltage (as this is easy to measure on the CRO) for different values of R and determine the internal resistance Ro and the voltage output εo of the signal generator. 1. Set up the circuit shown in figure 5. For R, use the variable resistance decade box. 2. Set the signal generator to 500 Hz and the sine wave amplitude control to maximum (keep the amplitude and frequency controls fixed throughout the measurements). 3. Increase R from 500 ohms to 5 kΩ in regular steps and measure the peak-to-peak voltage on the CRO. 4. Plot R/V versus R. From the slope and intercept of the best straight line find the values of Ro and εo (see equation 4). 5. Write a brief conclusion for this study. Figure 5 © KFUPM – PHYSICS revised 29/11/2007 49 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 MEASUREMENT OF e/m OF THE ELECTRON Purpose 1. To study the effect of electric and magnetic fields on charged particles and to measure the charge to mass ratio of an electron. 2. More practice in error analysis. Theory When an electron moves in a magnetic field B whose direction is perpendicular to the velocity v of the particle, it is acted on by a force F perpendicular to B and v with a magnitude given by: |F| = e v B (1) where e is magnitude of the charge of the electron. This force causes the particle to move in a circle in the plane perpendicular to the field. The force F is equal to the mass of the particle times the centripetal acceleration e v B = m v2 / r (2) where r is the radius of the circle and m is the mass of the electron. If the electron has been accelerated from rest through a potential difference V, then the kinetic energy is equal to the loss of potential energy. (1/2) m v2 = e V (3) Combining equations (2) and (3) gives e / m = ( 2 V / B2 r2) (4) Thus, when the accelerating voltage V, the magnetic field B, and the radius of the circular path r are known, the value of the e/m ratio can be computed and is given in units of Coulomb/kg by equation (4) if V is in volts, B in teslas and r in meters. Apparatus The Cathode ray tube used in this experiment is a special one in which the path of a beam of electrons can be observed directly. The tube is filled with argon gas at a pressure of 10-1 Pa. Electrons emitted by the heated cathode are accelerated by the potential difference applied between the cathode and anode cylinder. Some of the electrons come out in a narrow beam through a circular hole in the centre of the cylinder. This emission is then focused into a narrow beam by the grid of the tube. When electrons of sufficiently high kinetic energy leaving the cathode collide with argon atoms a fraction of the atoms will be ionised. Upon combination of these ions with stray electrons, the argon-arc spectrum is emitted and the characteristic blue colour of the undispersed visible light is observed. © KFUPM – PHYSICS revised 29/11/2007 50 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Figure 1 A homogenous magnetic field is produced in the region of the cathode ray tube by a current through two circular coils. Whenever a charged particle, such as an electron in the present experiment, is emitted, the force F given by equation (1) acts on the particle and will deflect it. The particle, therefore, moves under the influence of this force which has a constant magnitude but whose direction is always at right angle to the velocity of the particle. The orbit of the particle is, therefore, a circle. Calculation of the magnetic field The magnetic field produced at the position of the electron beam by a current I flowing through the coils must be calculated. For the equipment used in this experiment, there are two circular coils with N turns on each, the coils being connected to contribute equally to the field at the centre. This arrangement is known as a pair of Helmholtz Coils and has the advantage that the field is uniform in the region near the centre. For this particular geometry and constructional details of the apparatus, the relationship between B and I is given by: B = 0.716 µo I N / R (5) where µo is the permeability constant which is equal to 4 π x 10-7 Wb/A-m. , N = 130 turns and R is the radius of the coils. © KFUPM – PHYSICS revised 29/11/2007 51 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Procedure 1. Measure and record the diameter of the coils, then find the radius R. Phosphor coated crossbars Electron Paths Electron Gun r = 2 cm 3 cm 4cm 5 cm Figure 2 2. Use a regular magnetic compass to determine the plane of the earth's magnetic field at the location of the laboratory. Then set the equipment so that the plane of the Helmholtz Coils lies in the plane of the earth's magnetic field. Explain the importance of this step in your report. 3. The circuit has already been connected for you ( see Figure 1). The power supplies to the filament, coil, grid and anode are contained in one pack. Note the following, however. (i) The knob on the extreme right controls the anode voltage, U (0-500 volts). (ii) The second knob from the right controls the tube grid voltage. From the way the circuit is connected the grid voltage is negative. The grid voltage focuses the electrons into a narrow beam. (iii) The multimeter measures the accelerating voltage, V, which is the sum of the anode voltage and the magnitude of the grid voltage. © KFUPM – PHYSICS revised 29/11/2007 52 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals (iv) (v) (vi) PHYS102 The large demonstration ammeter measures the current, I, through the Helmholtz coils. The radius of the beam “r” is measured by allowing the beam to fall on one of the phosphor coated cross bars which allows the direct measurement of the radius, as shown in Figure 2 above. The CR Tube power supply uses high voltage (550 volts) and is capable of delivering high current (350 mA), which can be dangerous. 4. Switch on the power supply. This will automatically turn on the filament current. Allow one or two minutes for the filament to heat up. 5. Turn the anode knob to apply about 150 volts to the anode, and the grid knob to apply - 30 volts to the grid. (Note that the multimeter will read approximately 180 volts, and this is the accelerating voltage, V). A fine bluish-coloured beam will now be observed. 6. Switch on the current to the coils. Adjust the current and observe how the beam bends over and eventually circles back on itself. It may be necessary to rotate the whole tube to achieve this condition. 7. Set the current at some value, say 1.8 A. Adjust the anode voltage to give an accelerating voltage V between 150 and 250 volts so that the beam falls on the phosphor-coated crossbars. Before taking your measurements, however, make sure that with the current value chosen you will be able to bend the beam to fall on the 2 cm cross bar. If not, then increase the coil current slightly and decrease the anode voltage until the beam is observed to fall on the cross bar. Keep this value of current constant in the rest of the experiment. Note the radius of the orbit (2 cm) and the accelerating voltage, V. Adjust the anode voltage to get the circles of radii 3, 4 and 5 cm. 8. After finishing your measurements in step 7, do the following: (i) Turn the current through the coils to zero, reverse the current leads and increase the current. Observe that the beam is deflected in the opposite direction. Why does the beam deflect in the opposite direction? Comment on your observation in your lab report. (ii) Rotate the tube and observe the beam becomes a spiral. Explain why. Data analysis 1. Use equation (5) to calculate the magnetic field B. 2. Make use of equation (4) and your data, (V) and (r), in figuring out a way to graph the data so as to get a straight line. © KFUPM – PHYSICS revised 29/11/2007 53 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 3. From the slope of your graph and using equation (4), determine the value of e/m . 4. Estimate the errors in the radius r, voltage V and current I. 5. Calculate the error in r2, and plot error bars for r2 on your graph. 6. Find the maximum and minimum slopes and determine the uncertainty in the slope, then determine the uncertainty in the value of e/m . 7. Write down the value of e/m with the error to the correct number of significant figures. 8. The accepted value of e/m is 1.7x1011 Coul/kg. Calculate the percent difference in e/m value and discuss any discrepancy. © KFUPM – PHYSICS revised 29/11/2007 54 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 TANGENT GALVANOMETER Purpose 1. To study the tangent galvanometer. 2. To measure the horizontal component of the earth's magnetic field. 3. To find the magnitude of the earth's magnetic field. Background The earth exhibits a weak magnetic field at a given point on the earth's surface. This field has a horizontal component (parallel to the earth) and a vertical component (perpendicular to the earth). We will use an obsolete instrument, the tangent galvanometer, to measure the horizontal component of the field. We will then use a dip needle to measure the angle the total field makes with the horizontal. Knowledge of these two pieces of information will allow us to find the magnitude of the total field. Method A coil subjects a compass needle to a magnetic field Bc at right angles to the horizontal component of the earth's field Beh. The compass needle comes to rest along the resultant B = Beh + Bc at an angle θ from North, where tan θ = Bc / Beh © KFUPM – PHYSICS revised 29/11/2007 55 (1) Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 (hence the name tangent galvanometer). This equation allows us to find Beh if we measure Bc and θ. A dip needle is then used to measure the angle between the horizontal and the direction of the total magnetic field of the earth. Data 1. Carefully unscrew the entire compass case from the base. Measure and record d, the diameter of the coil. Remount the compass case. 2. Be sure that the compass needle is free to rotate. If not let down the device which protects the needle by means of the screw under the compass case. 3. Align the coil along the North-South axis using the compass needle as a guide. (This is something of a matter of judgment, but the eye is surprisingly good at these kinds of things.) Turn the compass needle case until the compass needle points to 90o, and the reading needle to 0o. Do not disturb the alignment of the instrument for the rest of the experiment. 4. Hook up the circuit below. Start with the N = 5 turns tap of the coil. You will find that the components of the circuit influence the compass needle slightly. Try to minimize this effect by keeping the tangent galvanometer at a distance from them. 5. With the rheostat set to maximum resistance, turn on the power supply and set the voltage to some value, say, 10.0 Volts. Vary the rheostat until a value of θ = 45o is obtained. Record the value of current. Repeat the procedure for the N = 10 and N = 15 taps. According to equation (1) this value of Bc is such that Beh = Bc . 6. Orient the dip needle provided along the direction of the earth's field (North-South direction) and measure the angle between the horizontal and the earth's total field. © KFUPM – PHYSICS revised 29/11/2007 56 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Calculations 1. Using the Biot-Savart law Bc = µoi 4π | dL x r | r3 you can show that the field at the centre of the coil is given by Bc = N µo I / 2 R (2) 2. Use this expression to calculate the value of Bc = Beh from the three values of current obtained above. 3. Average the three values of Beh. The accepted value of the horizontal component of the earth's field at Dhahran is 0.33x10-4 T. What is the percent difference between your value and this value? Is this error reasonable in view of the errors made in the various measurements? 4. Use the angle found in (6) above to calculate the magnitude of the earth's total field. Draw a diagram clearly showing the various components and angles used in this calculation. The earth's field is often estimated as 0.5 gauss. How does your values compare with this? © KFUPM – PHYSICS revised 29/11/2007 57 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 INTERNAL RESISTANCE AND EMF OF A CELL Purpose To measure the emf and internal resistance of a lead-acid cell. Background Every battery or power supply has an emf, ε, and an internal resistance, ri , which may be considered to be connected in series with it. The emf of a battery is the potential difference across its terminals when no current is drawn from it. When a current I is drawn from the battery, the terminal potential difference Vt decreases, and is given by Vt = ε - I ri . (1) Equation 1 shows that the terminal voltage Vt decreases with increasing current I. Therefore, if a voltmeter is connected across the terminals of a battery, the measured voltage is Vt and not ε. It is clear from equation 1 that Vt = ε only if I = 0; that is, no current is drawn from the battery. A potentiometer is a device to measure potential differences (voltage drops) accurately and at the same time no current is drawn from the battery (to the potentiometer). On the other hand, a voltmeter draws a current in order to measure a potential difference and thus changes the potential to be measured. In part A of this experiment the potentiometer is calibrated using a standard cell of known emf, εs, and is then used to measure directly the emf, εx, of a lead-acid cell. In part B, a digital multimeter (DVM) is used to measure the terminal voltage of the cell as a function of the current, I, flowing through it. The internal resistance, ri, can then be found from a graph of Vt versus I. - P. S + Ls or Lx O εs B A 15k G S1 Figure 1 © KFUPM – PHYSICS revised 29/11/2007 58 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 In Figure 1 the power supply (P.S.) maintains a constant potential difference between the points O and A of the slide wire. The voltage drop across point O and any other point B on the wire is therefore proportional to the length of the wire OB. A standard cell, whose emf, εs , is known is connected as shown in the figure. The tapping point B is moved along the wire until no current flows in the galvanometer, G. The potential drop between points O and B then equals the emf of the standard cell. Let the length of the wire be Ls; then εs is proportional to Ls. When the standard cell is replaced by the lead-acid cell whose emf is εx, and the new balance point is obtained, then εx is proportional to Lx. Hence, εx = ( Lx / Ls ) εs (2) In Figure 2, the terminal voltage Vt is measured as a function of the current, I. Note that the digital multimeter, DVM, has a high input resistance (~ 20 MΩ) and hence draws a negligible amount of current. DV ri εx S A Figure 2 Decade Resistance box Procedure. Part A. 1. Measurement of the emf of the lead-acid cell. Set up the circuit shown in Figure 1 and set the supply voltage to about 3 volts. Ask the instructor to check your circuit. Note that the 15-KΩ resistor is in the circuit to protect the standard cell from having large currents drawn from it when the tapping point B is far from balance, and also to protect the galvanometer from excessive current which could damage the pointer and the suspension. © KFUPM – PHYSICS revised 29/11/2007 59 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals 2. PHYS102 Adjust the position of the tapping point B on the slide wire until the current in the galvanometer is zero (null balance). Close the switch S1 and adjust the tapping point B to get a more accurate null balance. Record the length Ls. From now on, do not change the supply voltage. 3. Replace the standard cell with the lead-acid cell, and repeat step 2 to get a null balance. Record the length Lx. Part B. Measurement of the internal resistance of the cell. 4. Connect the circuit shown in Figure 2. Set the decade resistance box (0-10 Ω) at 10Ω. Decrease the resistance in the box in steps of 1.0 Ω (down to 1.0 Ω) and record the current and the voltage Vt shown on the DVM. Analysis 1. Using equation 2 and your values of Lx and Ls, calculate the emf, εx, of the lead-acid cell. The emf, εs, of the standard cell is 1.018 V. 2. Plot a straight line graph of Vt against I. 3. Using equation (1) and the slope of the graph, calculate the internal resistance, ri, of the lead-acid cell. 4. Using equation (1) and the y-intercept, calculate the emf, εx, of the lead-acid cell. Compare this value of εx with that determined in part A. 5. Estimate the error in εx and ri by plotting extreme lines through your data points and deduce new values of εx and ri from the intercept and slope. 6. State your values of εx and ri in the format x ± ∆x. Question A student uses a voltmeter (not a high resistance voltmeter) to measure the potential difference across the terminals of a battery. Is he measuring the emf of the battery? Explain. © KFUPM – PHYSICS revised 29/11/2007 60 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 KIRCHHOFF'S LAWS Purpose To study Kirchhoff's laws in the case of a two-loop circuit. Theory Consider the two-loop circuit shown in Figure 1. There are two Kirchhoff's laws. The first says that at any junction the algebraic sum of the currents must be zero. (Note: We arbitrarily label positive a current approaching a junction and negative a current leaving it). For example, in the circuit of Figure 1, at junction 2 we have: I1 + I2 + I3 = 0 (1) The second Kirchhoff's law says the algebraic sum of the changes in potential around a loop equals zero. For loop 1 above we have E1 - I1 R1 + I2R2 - E2 = 0 (2) For loop 2 E2 - I2R2 + I3R3 = 0 (3) Equations 1, 2 and 3 can be re-written as: I1 + I2 + I3 = 0 R1 I1 - R2 I2 = E1 - E2 R2 I2 - R3I3 = E2 © KFUPM – PHYSICS revised 29/11/2007 61 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 The solutions for I1 , I2 and I3 are: I1 = [R2E2 + (R2 + I2 = [R1E2 I3 = - [(R1 + R3)(E1 - E2)] / D (4) R3(E1 - E2)] / D R2) E2 + R2 ( E1 - (5) E2 ) ] / D (6) where D = R1R2 + R2R3 + R1R3 (7) Procedure The electrical circuit used in this experiment is shown in Figure 1. It includes two emfs (one power supply and a flashlight battery), and three resistors, connected in a two-loop circuit. 1. Set up the circuit shown in Figure 1, paying special attention to the polarity of the two emfs. Notes: (a) Your instructor will assign the values of R1, R2 and R3 from the 2*3=6 combinations of three resistors (27 Ω, 47 Ω and 68 Ω) which are available. (b) You will measure voltages and currents using a digital multimeter. 2. Close the switches. Turn on the power supply. Turn the knob until E1 = 10.0 V, as measured by the multimeter in the voltmeter function. Use the multimeter to measure E2. 3. At junction 2 remove the wires from resistors R1, R2 and R3, one at a time, and measure the currents I1, I2, and I3, using the multimeter in the ammeter function. 4. Measure the voltage V1, V2, and V3, across R1, R2 and R3, respectively. Note: Use the switches to temporarily disconnect the emf's from the circuit while you are calculating, etc. Otherwise the battery will run down, making your measurements vary with time. When measuring the currents note that the current is positive if the ammeter deflects upscale when the "COM" input is connected to junction 2 and the V-Ω-A input is connected to the resistor; otherwise switch leads, measure the current and label it negative (see Figure 2). © KFUPM – PHYSICS revised 29/11/2007 62 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 i i 1 1 2 com positive current Com 2 V-Ω-A negative current V-Ω-A Figure 2. How to determine the direction of the current The sign of the voltages is determined as shown in Figure 3. com i V-Ω-A i com V positive V-Ω-A V negative Figure 3. How to determine the sign of the Analysis 1. Substitute the values of E1, E2, R1, R2 and R3 into equations 4, 5, 6 and 7, and obtain the value of I1, I2 and I3. 2. Using the calculated values of I1, I2 and I3, calculate V1, V2 and V3 using Ohm's law. Questions 1. Does the sum of the measured currents at junction 2 equals zero in accord with the first Kirchhoff's law? If not, can you explain the difference? 2. Are the sums of the measured changes in potential around loop 1 and around loop 2 equal to zero in accord with the second Kirchhoff's law? Can you explain any differences? 3. How well do the measured values of I1, I2 and I3 agree with the calculated values in magnitude and sign? Is the error reasonable considering the measurement accuracy and the resistor tolerance? 4. What is the first Kirchhoff's law equation for junction 1? Does it yield any new information about the current? Explain. © KFUPM – PHYSICS revised 29/11/2007 63 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 INTRODUCTION TO ELECTRICAL MEASUREMENTS Purpose To teach freshman students how to 1. operate basic electrical measuring instruments such as galvanometer, ammeter and voltmeter, 2. use an analog multimeter / avometer, a digital multimeter, and an ohmeter, 3. assess the precision of each of these measuring instruments, 4. set up a simple one-loop DC circuit and measure some of its characteristics, 5. use an oscilloscope to measure the voltage as well as the frequency of alternative signals. Equipment 1. Galvanometer, Ammeter, Voltmeter. 2. Avometer, Digital Multimeter. 3. DC power supply, resistors, switches, connecting cables. 4. Dual-trace oscilloscope, signal generator (sinusoidal and square signals), coaxial cables. Note: This laboratory "experiment" consists only of demonstrations by the laboratory instructor followed by practice work by the students. No lab report needs to be written. © KFUPM – PHYSICS revised 29/11/2007 64 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 MEASUREMENT OF THICKNESS OF A VERY THIN SAMPLE Purpose To estimate the number of atoms in the thickness of a pencil line. Background Consider a very thin strip of a conducting material (Fig. 1). Its thickness, t, is much less than a micron (micrometer, 10-6 m) . It is not possible to measure its thickness, even if we try to use a micrometer. Can you guess why? L t W Multimeter Figure 1. Measurement of resistance of a thin metal sheet. Measurement of thickness of such a thin sample has generally been made by a use of an indirect method. One of these methods is outlined below. The resistance, R, of a conducting rod is given by L R = ρ …………………….……………..(1) A where ρ is the resistivity, L is the length and A is the area of cross-section of the rod. If we have a conductor in the shape of a strip, as shown in Fig. 1, then its resistance, R, measured along the length is given by R=ρ L L =ρ …………………………….(2) A Wt where ρ is the resistivity of the strip, L, W and t are shown in Figure 1. The thickness, t, of the strip can be determined from Eq. (2), if we measure R, L and W, and use the known value (provided, it is available) of the resistivity ρ, of the strip material. Resistivities of some pure materials, such as Cu, Al, Au, C, etc. are given in “PHYSICS” by SERWAY. However, for different alloys and impure materials, the resistivities are not known. In such cases we have to determine ρ first. © KFUPM – PHYSICS revised 29/11/2007 65 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 L d Multimeter Figure 2. Measurement of resistance of pencil lead. Study (1) Measurement of resistivity of a pencil lead. 1. Measure the length, L, and diameter, d , of the pencil-lead (Fig. 2) 2. Using a multimeter, measure the resistance, R, of the pencil lead across its length (Figure 2). 3. Using the above data and Eq. (1), calculate the resistivity, ρ, of the pencil lead. Study (2) To estimate the thickness of a pencil line and the number of atoms in the thickness of the pencil line. 1. On a sheet of graph paper, using a 4B pencil, draw a line 2.0 mm wide and 160 mm long with a 10 mm square at one end. Shade in the line and square so that they are dense black (see Fig. 3). 10 mm 2 mm 10 mm l 160 mm x=0 Fig. 3 Measurement of resistance Multimeterat different distances (measured from x=0) along the pencil line. The square (Figure 3) is to act as an electrical contact at the zero end of the line. © KFUPM – PHYSICS revised 29/11/2007 66 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 2. Keep one connector from the multimeter firmly pressed in the square and press the other firmly at different distances along the pencil line (Fig. 3) in order to find the resistance, R, for several lengths A of the pencil. Record all your results in a table. 3. Plot a graph of R against A , and draw a straight line of best fit. 4. Find the slope of the straight line. Use this value together with the known values of ρ (found in study 1) and W (= 2mm), to find the thickness, t, of the pencil line, from Eq. (2). 5. Assuming the size of an atom of carbon in the pencil lead to be 2 x 10-10 m, estimate how many atoms thick your pencil line is. © KFUPM – PHYSICS revised 29/11/2007 67 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Cal Lab Simulation of Standing Waves on a String A step by step, quick tutorial on how to make sine waves in EXCEL, make them travel and make them ‘stand’ by summing two traveling waves! Background Assume that we have a string two meters long fixed at both ends (as in your experiment “Standing Waves on a String”) with the frequency of the vibrator given by f = ω/2π with ω = 10 rad/sec. Enough mass is hung to make it resonate in four segments (n = 4). The wave function for the waves traveling to the right will be given by Y1 = A sin(kx - ωt), and the wave function for the waves traveling to the left will be given by: Y2 = A sin(kx + ωt). The wave function for the resulting standing waves will be given by: Y = Y1 + Y2 = 2A sin(kx) cos (ωt) = B(x) cos (ωt) In this case, at x = L , B(L) = 0 , k = 2π / λ, then kL = nπ and λ = 2L/n , where n = 1, 2, 3, 4,… In our case: n = 4, L = 2 m, λ = 1 m, k = 2π and ω = 10 rad/sec. Assume that the wave amplitude is given by A = 0.03 m. [Note that the value of π = 4*Atan (1), since tan (45o) = tan (π/4) = 1, hence π = Pi = 4*Atan (1).] To simulate the situation using MS Excel, we shall enter as inputs the following constants and formula: Pi k A ∆x ∆t ω t x Y1 Y2 Y = = = = = = = = = = = © KFUPM – PHYSICS revised 29/11/2007 4*Atan(1) 2*Pi 0.03 m Delta x = 0.005 m Delta t = 0.01 sec Omega = 10 rad/sec 0 sec (for a start) 0 m (for a start) A sin(k x - ωt) A sin(k x + ωt) Y1+Y2 68 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Simulation 1. Enter the above as shown in Table 1 A 1 2 3 4 5 6 7 8 9 10 11 12 13 B C D Pi= k= A= Delta x= Delta t Omega = =4*Atan(1) =2*Pi 0.03 0.005 0.01 10 (1/m) m m sec Rad/sec E F G t= 0 (sec) X (m) y1 (m) y2 (m) Y=Y1+y2 (m) 0 =A*Sin(k*B12-Omega*t)=A*Sin(k*B12+Omega*t)=C12+D12 =B12+Delta_x =A*Sin(k*B13-Omega*t) =A*Sin(k*B13+Omega*t) =C13+D13 Table 1 Then you will see Figure 1. Figure 1 2) Define names as shown in Table 2 Click on the cell C3 © KFUPM – PHYSICS revised 29/11/2007 Click on name box Type then See Figure 1 Enter key C3 Pi 69 hit See Figure 2 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals C4 C5 C6 C7 C8 F9 C4 C5 C6 C7 C8 F9 PHYS102 k A Delta_x Delta_t Omega t Table 2 Figure 2 When you have finished you will see Figure 3. Figure 3 © KFUPM – PHYSICS revised 29/11/2007 70 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals 3) PHYS102 Select from column B13 → E13 and down on rows until the sign reading 400 rows by 4 columns appears {400R x 4C}, then stop. See Figure 4 below. 400Rx4C Figure 4 4) #5 5) From Menu choose Edit → Fill → Down. Then click. Then you will see Figure Figure 5 a) Select from B12 → E12 and down ward to B412 → E412. © KFUPM – PHYSICS revised 29/11/2007 71 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 b) Click on Chart and select xy (scatter) as shown in Figure#6. Then click Finish Figure 6 You will now see figure 7. Y1 and Y2 will be on top of each other. This is a snap shot at t = 0. 0.08 0.06 0.04 0.02 0 -0.02 0 Series1 Series2 1 2 3 Series3 -0.04 -0.06 -0.08 Figure 7 © KFUPM – PHYSICS revised 29/11/2007 72 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals 6) PHYS102 a) Double click on X-axis values. Then click on Format axis. You will see Figure 8. Choose scale. Uncheck where the arrows and fill with the correct values, then click OK. Figure 8 b) Do the same as on a) for Y-axis values. Then you will see Figure 9. Uncheck where the arrows point and click OK. Figure #9 © KFUPM – PHYSICS revised 29/11/2007 73 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 c) Right click on legend. Then click on Format legend. Choose Placement and you will see Figure 10 . Choose location for legend as shown in Figure 10 and click OK. Figure 10 d) Click on chart area. From Menu choose Chart → Source of Data. Choose Series you will see figure #11.Enter names such as Y1 ,Y2 and Y then click OK. Figure 11 e) Click Plot Area. From Menu choose Chart → Chart Options. © KFUPM – PHYSICS revised 29/11/2007 74 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 You will see Figure 12. Figure # 12 Fill as shown in figure #12 then click OK. You will see Figure # 13 Figure # 13 You are now ready to Animate the Waves 7) a) From Menu choose Tools → Options. You will see Figure 14. © KFUPM – PHYSICS revised 29/11/2007 75 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Figure 14 Choose calculation and you will see Figure 15 Figure 15 © KFUPM – PHYSICS revised 29/11/2007 76 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Fill According to arrows shown in figure # 15 then Click OK. b) Type =F9 + Delta_t in F9 column as shown in Table #3 A 1 2 3 4 5 6 7 8 9 10 11 B C Pi= k= A= Delta_x= Delta_t Omega= =4*Atan(1) =2*Pi 0.03 0.005 0.01 10 x D y1 y2 E F t= =F9 +Delta_t Y=Y1+y2 Table 3 c) Now press continuously F9 key and you will see the waves animated, Y1 moving to the right, Y2 moving to the left, and Y is standing (not traveling to the right or to the left). Figure 16 is a snap shot at t ≠ 0. Figure 6 Congratulations! You are done. To restart iterations from 0 type 0 in the F9 column. Then repeat step 7 b) and c). © KFUPM – PHYSICS revised 29/11/2007 77 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Appendix A THE LABORATORY BAROMETER The instructor will demonstrate the procedures to be followed in using the barometer to determine the atmospheric pressure. There are four main steps: 1. Zero the barometer by turning the levelling screw until the pointed tip of the ivory peg just touches the mercury surface. 2. Adjust the vernier slider until the bottom of the slider is level with the top of the mercury column. Read the main scale of the barometer to the nearest mm. 3. Read the vernier scale to the nearest 1/10 mm and add this reading to the main scale reading. Note: © KFUPM – PHYSICS revised 29/11/2007 Refer to the next page for examples of vernier scale readings. 4. Read the temperature on the barometer and subtract the 'temperature correction'. Example: If the main scale reading is 754 mm, the vernier scale correction is 0.3 mm, and the barometer temperature is 22o C, what is the corrected barometric (atmospheric) pressure? Answer: Atmos. pres. = 754 mm + 0.3 mm - 2.7 mm = 751.6 mm Hg (or torr) 78 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 Vernier scale readings Check that you agree with the readings shown below: © KFUPM – PHYSICS revised 29/11/2007 79 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals PHYS102 TEMPERATURE CORRECTION FOR BAROMETRIC READINGS Temperature Barometer reading (mm of Hg) oC ----------------------------------------------------------------------------------------------------725 730 735 740 745 750 755 760 765 14 1.7 1.7 1.7 1.7 1.7 1.7 1.7 1.7 1.7 15 1.7 1.7 1.7 1.8 1.8 1.8 1.9 1.9 1.9 16 1.9 1.9 1.9 1.9 1.9 2.0 2.0 2.0 2.0 17 2.0 2.0 2.0 2.0 2.1 2.1 2.1 2.1 2.1 18 2.1 2.1 2.2 2.2 2.2 2.2 2.2 2.2 2.2 19 2.2 2.2 2.3 2.3 2.3 2.3 2.3 2.4 2.4 20 2.4 2.4 2.4 2.4 2.4 2.4 2.5 2.5 2.5 21 2.5 2.5 2.5 2.5 2.5 2.6 2.6 2.6 2.6 22 2.6 2.6 2.6 2.6 2.7 2.7 2.7 2.7 2.7 23 2.7 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.9 24 2.8 2.8 2.9 2.9 2.9 2.9 3.0 3.0 3.0 25 3.0 3.0 3.0 3.0 3.0 3.0 3.1 3.1 3.1 26 3.1 3.1 3.1 3.1 3.1 3.2 3.2 3.2 3.2 27 3.2 3.2 3.2 3.2 3.3 3.3 3.3 3.3 3.4 28 3.3 3.3 3.3 3.4 3.4 3.4 3.4 3.5 3.5 29 3.4 3.5 3.5 3.5 3.5 3.5 3.6 3.6 3.6 30 3.5 3.6 3.6 3.6 3.6 3.7 3.7 3.7 3.7 Note: SUBTRACT the appropriate correction factor to allow for differences in expansion of mercury in the brass barometer at different temperatures. Example: If room temperature is 25.3o C and the barometer reading is 746.7 mm, then from the table above, the appropriate correction factor is 3.0 mm and the corrected barometer reading will be 743.7 mm of Hg. © KFUPM – PHYSICS revised 29/11/2007 80 Department of Physics Dhahran 31261 King Fahd University of Petroleum & Minerals © KFUPM – PHYSICS revised 29/11/2007 PHYS102 81 Department of Physics Dhahran 31261