About half has past… What have we learned: 1. Kinematics – description of motion 2. Dynamics – reasons for motion 3. Applications to some physical phenomena Now: Electricity – just one more force 1. What property is necessary for this force to act? 2. Description of the force… 3. What does this force depend on? Example – force of gravity: 1. Any massive object (that has a property of gravitational mass) creates the gravitational field around it. (The mass changes the space.) If another massive object appears in this “changed” space, a gravitational force acts on it. 2. The force of gravity between two point-like objects is central, attractive, and proportional to both masses. 3. The force depends on the distance between the objects, ~ 1/d2. EXPERIMENTS What have we learned ? • When we rub objects they acquire a ‘charge’. • Like charges repel, unlike charges attract. • Force between charges depends on the distance What sort of property is “charge” ? Charge is a property of some of the subatomic particles that “make up” matter. Charge comes in two varieties, by convention, POSITIVE and NEGATIVE By convention: electrons carry negative charge protons carry positive charge neutrons carry no charge – they are neutral What happens when you rub a piece of glass/rubber ? • Charges can neither be created nor destroyed. • They can be “separated” – in this case negatively charged electrons leave the glass charging the silk negatively and the glass positively (fur positively, rubber negatively). • The net charge remains zero as it was in the beginning. CONSERVATION OF CHARGE: Total charge can neither be created nor destroyed. net charge = positive charge – negative charge Example: you connect three charges together +3nC, +5nC, and -4nC, the net charge is: 3 + 5 - 4 = 4nC When I rub a piece of rubber, some charges (electrons) get removed from the fur and transferred to the rubber. IONS are formed: ION = charged atom Negative ION Positive ION Electric force vs. Gravitation Electric forces can be repelling or attractive: e.g. two positive charges repel each other, a positive and a negative charge attract. Electric forces are much stronger than gravity. The pith ball did not move unless is was charged. If the charges of two particles is zero, the electric force between them is zero. Gravity acts on everything that is massive (no negative mass). Why are electrical forces important ? • All atomic and molecular interactions, and therefore structure of all atomic matter is guided by electric forces • Most of (macroscopic) forces that surround us in everyday are electric in nature: string tension, elastic forces of springs, any solid or liquid, friction in solids and liquids, etc. • Most of the devices use electric power, or electrically guided… UNITS: Charge is measured in Coulombs (C) The charge of one electron is only 1.6 x 10-19 C 1 Coulomb is a very large unit – not really convenient for electrostatics The reason: 1 C/s = 1 Ampere COULOMB’S LAW Two ‘point’ charges q1 and q2 act on each other with the electrical force proportional to these charges and inversely proportional to the distance squared between them. q1 q 2 F = ke 2 , d N⋅m k e = 9 × 10 2 C 9 + 2 d + The force is acting along the straight line connecting the charges and is attractive if negative and repulsive if positive. What is the force acting between two +1 µC charges 1 cm apart? −6 −6 10 × 10 9 F = 9 × 10 = 90 N −2 2 (10 ) q1q2 F =k 2 d Match the picture the with force acting on the left charge in each case: 1 + 2 3 4 + +- + 5 - 6 - 7 - 8 + + What is the force on the charge on the right ? 1. 2. 3. Zero 4. None of the above Answer 2 – because of the Newton’s 3rd law QUESTION: In which of the two situations shown is the electrical force larger ? Or are they the same ? q q + + d 2 q qq F = ke 2 = ke 2 d d 2q ++ q 2d + 2 2 2qq 2q 1 q = ke = ke 2 F = ke 2 2 2 d ( 2d ) 4d What does a charge do to make other charges feel a force ? An electric charge alters space around it by creating the electric field. Another charge in this field is acted upon by the electric force. Electric (static) force: 1. Any charged object (that has a property of electric charge) creates the electric field around it. (The charge changes the space.) If another charged object appears in this “changed” space, an electric force acts on it. 2. The electric force between two point-like charged objects is central, attractive or repulsive, and proportional to both charges. Attractive between opposite charges and repulsive between like ones. 3. The force depends on the distance between the objects, ~ 1/d2. Electric field (strength) Force acting on a charged object Electric field strength = charge on this object Example : Find the force acting on the F electron in an electric field of 100 N/C. Electric field E = q E = F ⇒ F = q × E = −1.6 × 10 −19 × 100 q N Unit : C = − 1.6 × 10 −17 N 1.6 × 10 −17 F Find acceleration : a = = − m 9.1 × 10 −31 13 m = 1.8 × 10 2 s The electric potential energy Imagine you have two opposite charges some distance apart from each other. Let’s fix one of the charges. One can apply an external force to a movable charge and move it to a larger distance from the fixed charge. The work done by the external force will be positive because it is directed in the same direction as the displacement. This force will “work” against the electric force and the latter will do negative work. After that, one can release the charge, and it will be attracted back to the fixed charge by the electric force. The electric force will then do positive work. We can introduce the potential energy in the same fashion we did for the gravitational field The external force does positive work against the electric force, the potential electric energy increases by exactly this amount. Then, when the charge is allowed to move, the electric force does positive work, and the potential energy decreases. Potential difference This potential energy (and the work of the external force) is proportional to the product of these charges. Then (similarly to the introduction of the field) we can divide the potential energy by the value of the moving charge and introduce a new scalar quantity, the electric potential. This quantity is independent of the moving charge and characterizes the electric field of the fixed charge. The potential as well as the potential energy is a relative term – you have to choose the point of zero potential. Then the electric potential at any other point will be a potential difference between that point and the chosen point of zero potential. This is similar to what we did with gravitational potential energy ∆PE = mgd d – is the vertical distance between two positions – a relative term Sometimes it is useful to introduce an “absolute” value for potential (and therefore PE), namely – zero at infinity where the field is zero. The sign convention is such that if an object is charged positively, the potential around it is positive, negatively – negative. One can think of it as a potential to attract (if negative) or repel (if positive). The same sign convention is valid for PE. Total energy and bound states • If we ascribe zero potential to infinity, and there is a positively charged object (a nucleus), then negatively charged objects will be attracted to it. • If this negatively charged object rotates around the positively charged one on a stationary orbit – such that the centripetal force is supplied by the electric attraction. This forms a bound state. • The total energy (KE+PE) of the bound particle is less than zero. Therefore the charge cannot escape (as the planet cannot), because otherwise at infinity where PE=0, the KE would have to be negative which is impossible. Energy 0 Unbound states E>0 Distance Maximum separations One of the bound states E<0 Conductors and Dielectrics Materials can be divided into two groups - conductors and dielectrics. In dielectrics, electrons are tightly bound to their hosts, the atoms or molecules keeping them neutral. If such material is exposed to the electric field, the molecules or atoms may rearrange themselves a little (the shape of electron clouds changes) to “make the negative charges closer to the positive charges”. However, free motion of charges are not possible. In conductors, some electrons are “free” from their hosts and can readily move when the field is applied. Typical conductors – metals, salty water… Typical insulators – plastics, ceramics, etc. QUESTION: Can there be electrical forces between a charge object and a neutral object ? 1. Yes 2. No Let’s do the experiment… Why ? “Neutral” objects also contain charges. They just have the same amount of positive & negative charge, balancing each other. If charges can move, objects can become ‘polarized’. Three different ways of charging objects: 1. Charging by contact – in this case as you touch the object, the electrons rearrange: a. If you have a negatively charged object – there is an “excess” of electrons – some of them (but not all) move to the other object and charge it negatively. b. If you have a positively charged object – there is a lack of electrons – some of the other object’s electrons partially recover this deficit and charge the other object positively. In both cases the charge of the object that you charge is the same as the charge of the originally charged object 2. Charging by induction: Only for conductors – a. Bring a charged object close to the conducting object, but not touching it. b. Connect the conducting object with the ground c. Remove connection with the ground d. Remove the originally charged object a. The opposite charges are attracted to be closer to the charged object, and the like charges as far as possible b. The like charges can now escape c. Now they cannot return d. The object “lost” like charges and is therefore charged with charge opposite to the original. 3. Charging by rubbing – in this case the charges are separated as a result of mechanical work. • Charges of two objects are equal but opposite • What object acquires negative charge? – The one whose affinity to electrons is larger – balloon and hair, plastic and fur, silk and glass, etc. Charges can move. The motion of charges is called the electric current CURRENT = amount of charge that flows by a location in space per time. I = q/t Unit: Ampere = 1 A = 1 C/s “Amps” Direction of the electric current By convention, the positive direction of the current is the direction of motion of a positive charge which is repelled by positive charges and is attracted by negative charges. If a loose positive charge is located at a position with a higher potential, it will move in the direction of the lower potential, so that the electric force does positive work and the charge loses the potential energy as it is moving. Since the negatively charged electrons are the carriers of charge in any metal conductor, they move in the opposite direction to the conventional direction of the current; e.g. if the (positive) current is flowing from A to B in a copper wire, it means that the electrons are drifting from B to A. If charges move – electric current = charge passing by per unit time I = q/t , Units: Ampere = Coulomb/second Conventional direction of current from higher potential to lower potential regardless of the charge carrier – compare the result of two events 1. Positive q moved from left to right 2. Negative q moved from right to left + I - For the charge counter that counts charges to determine the current, q/t, there is no difference whether positive moved to the right or negative moved to the left Small drift – large current Let’s say we have a current of 1A in copper wire of cross section area of 1 mm2 at a room temperature. What is the thermal average speed of the electrons? It’s 1.2×105m/s. If all electrons were moving with this speed along the wire, the current would have been 3.2×105A !! Instead they rapidly move back and forth with this speed and very slowly drift with a drift velocity which is about 4 mm/s in this case. Therefore, the electrons do not have to move much on the average to cause a substantial current. Constant force – constant current? 1. If a charged particle is placed in a uniform field E, a force F = qE is acting on it, and its acceleration = F/m 2. What happens if a conductive wire is placed in the electric field that is “maintained” constant by some external means? – A constant (direct) electric current is the result. 3. Where is the acceleration? Electrons accelerate, but collide with the lattice or other electrons, stop, then accelerate again, and so on. On the average, they move at a constant speed making the current constant. 4. This drag – is the reason for resistance – “the opposition” to the current flow. 5. What does the resistance depend on? RESISTANCE – a measure of the opposition to current flow In regular conductors like metals it depends on 4 factors: 1. Properties of the material – conductivity the higher the conductivity, the lower the resistance. 2. The length of a conductor – the longer the wire, the larger the resistance 3. The cross section area of the conductor – the larger the area, the lower the resistance 4. The temperature – the higher the temperature, the larger the resistance UNITS: OHMS (Ω) 1 l R= σ A 1 σ = l σ0 (1 + β∆T ) 1. Can use the above formula and reasoning 2. Totally different picture – electrons accelerate in the field and barely collide with anything – cannot use the formula… 3. More difficult dependence than in (1) because of ion motion. The formula is not valid Voltage in a circuit The current is flowing through the circuit because the carriers of charge can move and because there is a potential difference between the terminals of the circuit. This potential difference is due to a work done by the external forces in a battery or a power supply. This work is done against the electric force and it results in separation of charges, or in other words, charges gaining potential energy. This potential energy is equal to the amount of separated charges multiplied by the potential difference across the power supply or voltage across the power supply: W external = PE = qV . You will see that it is much more convenient to discuss this voltage across the power supply rather than potential energy because voltage is independent of charge and is an adequate characteristics of a power supply. Analogy – if the current that you observe is similar to a waterfall, the battery is similar to the water pump that pumps the water upwards. V = Work/q = PE/q UNIT: 1 Volt = 1 V = 1 J/C Now, after being separated the positive and negative charges can happily reunite, but not through the battery – through the circuit – and by doing that they will spend the acquired energy. Electro – mechanical analogy How an electron spends its energy − e − e When a charge moves along a wire moved by the electric force (field), the electric force does work – the charge loses its potential energy in the electric field. Where does it go? to going through resistance! – to internal energy, light, etc. 1. On each part of the circuit, some energy is spent, therefore some voltage is reduced as well. 2. The voltage drop across a portion of the circuit is proportional to the current and the resistance of this portion – Ohm’s law V = I×R, V I= , R V R= I 3. Alternatively: The current in the circuit is equal to the voltage across it divided by its resistance. 4. Or: The resistance of a circuit element is equal to the voltage drop on it divided by the current in the circuit. Example: A 100 Ω light bulb is connected to a 9 V battery. How high is the current ? V=IR I = V / R = (9V) / (100 Ω) = 0.09 A = 90 mA NOTE: 1 Ω = 1 V/A Example: You are making a circuit from a 12V battery and a resistor (an element that has a calibrated resistance). If you want a current to be less than 1mA, what resistance should you use? V = I×R, V I= , R V R= I 12V R= = 12,000Ω = 12kΩ 0.001A Example: What voltage is needed to get a 10A current through the 20Ω resistance? V V = I×R, I= , R V = 10A × 20Ω = 200V V R= I CIRCUITS SERIES CIRCUIT PARALLEL CIRCUIT QUESTION: A 10 Ω light bulb is connected to a 12 V battery. Compare this situation to a circuit where three 10 W bulbs are connected to a 12 V battery in series. What can you say about the series circuit? 1. The resistance of the circuit. 2. The voltage from the battery. 1. Is increased 3. The current. 2. Is the same 4. The voltage on each bulb. 3. Is decreased 5. The brightness of the bulb. QUESTION: A 10 Ω light bulb is connected to a 12 V battery. Compare this situation to a circuit where three 10 W bulbs are connected to a 12 V battery in parallel. What can you say about the series circuit? 1. The resistance of the circuit. 2. The voltage across the battery. 1. Is increased 3. The current. 2. Is the same 4. The voltage drop on each bulb. 3. Is decreased 5. The brightness of the bulb. 6. The current through each bulb Now we know that charges spend their energy traveling along the circuit. Apparently we use some of this energy. Consider a light bulb – let’s call the energy used by this bulb useful (regardless of the efficiency) When a charge q passes through the bulb, the charge loses energy E=qV, where V is a voltage drop across the bulb There are many charges pass the bulb over time t. How many? – The rate is q/t = I, therefore q = It Then E = VIt – the energy dissipated on a bulb over time t. Can also introduce the power dissipated on the bulb P = E/t = VI ELECTRICAL POWER V = Energy per unit charge = E/q I = Charge per unit time = q/t Power = Energy per unit time = E/t = (Energy per unit charge) x (Total charge per unit time) P=VI Units of power = Volt x Ampere = J/C x C/s = J/s = Watt QUESTION: A 100 W light bulb is connected to 100 V power supply. What is the current through the bulb ? P = V I ⇒ I = P/V = (100W)/(100V) = 1 A Power and Ohm’s law V I= , R V V = I×R, R= I 2 V 2 P = VI = I R = R If you want to increase the power dissipated on, say, light bulb, what should you do? • Increase its resistance? • Decrease its resistance? QUESTION: An automobile headlight and a dashboard light use the same voltage, but the power input to the headlight is much larger because 1. Its resistance is lower 2. The current in it is larger 3. It uses energy faster 4. All of the above 9P=V2/R, I=V/R, P A searchlight installed on a truck requires 60 watts of power when connected to 12 volts. a) What is the current that flows in the searchlight? b) What is its resistance? ANSWER: a) P = VI 60W = 12 V x I 60W/12 V = I I=5A B) V = IR 12 V/5 A = R R = 2.4 Ohms Example: A light bulb is rated at 60W when connected to 120 V. • What current flows through the bulb in this case? • What is the bulb’s resistance? • What would be the current if it were connected to 60V (if the resistance is the same)? • What would be the power consumption in this case? 1. I = P / V = 60 / 120 = 0.5A 2. R = V/I = 120 / 0.5 = 240Ω 3. I = V/R = 60 / 240 = 0.25A 4. P = VI = 60 ⋅ 0.25 = 15W kW-h – kilo Watt hours = unit for the consumed energy Example: The toaster is rated 1,500 W. It takes 6 min to make a toast. How much does it cost if the energy price is $.1/kW-h ? 6 1. Energy consumed = 1.5kW ⋅ hour = 0.15kW ⋅ h 60 2. Cost = .1$/kW ⋅ h ⋅ 0.15kW ⋅ h = $0.015 The resistance of an electric heater is 10Ω when connected to 120 V. How much energy does it use during 30 min of operation? 2 2 V 120 E = Pt = t= W ⋅ 0.5h = 0.72kW ⋅ h R 10 AC = Alternating Current • Household outlets • Used for lights, heaters etc. where direction of current doesn’t matter. • Can be stepped up or down by using transformers. DC = Direct Current •Can be produced from AC by “rectification” • Needed for electronics etc. • Batteries produce DC voltages Some more things that I want to mention Semiconductors – are close to insulators – have a complete band of electrons, but if something changes – temperature or impurities are added, they may acquire free carriers of electric charge intrinsic p – type n - type Plasma – the fourth state of matter – all (or most) atoms or molecules are ionized • the temperature is high enough so the kinetic energy of electrons is larger than ionization energy • dense and rarefied plasmas • gas with two temperatures – ion and electron • Electron–positron plasma, fusion, etc. Piezoelectric Effect Crystals which are capable of separating charges when compressed, twisted or distorted are said to be piezoelectric. Mechanical work is used to separate charges. The voltage due to these separated charges can be used for the external loads such as light bulbs in shoes or the speakers in the vinyl disc players. Many applications in technology… MAGNETISM First we approach to the magnetic force as just another force Then we will notice that the magnetic force is related to the electric force, and that the electric and magnetic fields are related as well We will discuss the features of the electromagnetic field and will notice problems that cannot be resolved in frames of the Newtonian physics and require the new physics, the theory of relativity. First notion: Magnets Have two poles (North & South) Poles cannot be separated Like repel, unlike attract Magnets create a field around them: The Earth’s Magnetic field Does the compass needle always point towards the north pole of the Earth? No, the magnetic north pole is not the same as the geographic north pole. QUESTION: Which of the following statements is true: 1. The north pole of a compass needle points toward the north pole of earth. 2. The south pole of a compass needle points towards the north pole of earth. The first statement is correct, but it’s all about conventions: The south magnetic pole of the Earth is near the north pole of the Earth N What is magnetism ? What we know so far: • Electric charges produce electric fields around them. •An electric field causes a force on any charged object placed in it. •Magnets produce magnetic fields. •A magnetic field causes a force on the poles of any magnet placed in it. It turns out that electricity and magnetism are related Experiment with moving charges – electric current and a compass A moving electric charge (current) produces a magnetic field around it. Thus magnetic fields is caused by the electric current… The magnetic field around a wire with a current is proportional to the current and inversely proportional to the distance from the wire. Electromagnet Earth’s magnetic field I B~ r Why are some materials “magnetic” and others are not? Electrons move around atoms = Moving charge! Electrons produce their own tiny magnetic field. In some materials these fields can all line up, producing a big field. These materials are called “Ferromagnetic”. Moving charge (current) in a magnetic field: A magnetic field exerts a force on a moving electric charge (current) The force on the moving electric charge (current) is proportional to the charge, its velocity, and the strength of the field v F = km q B c F = IB∆l Ampere Lorentz force force The forces in both cases are perpendicular to the both magnetic field and velocity of the charge The first problem with Newtonian mechanics Have you noticed that the force is proportional to the velocity of charge? There is a problem here: think of different observers – recall that two observers in different inertial frames (moving with respect to each other uniformly) do not agree about the velocity of the moving particle but agree about its acceleration and the force acting on it! So if the force explicitly depends on the velocity, it may be zero in one inertial frame and not zero in another! The observers disagree about the Force! This problem cannot be resolved within Newtonian mechanics. We will remember this (and everything else I’ve ever said) and continue our study of the magnetic field. Side note Some of you may have noticed that we have already seen a force that depended on the velocity – the force of air resistance. It did not cause any agitation. Why? Because the air resistance is due to collisions with molecules, similar to the pressure force on the walls of a vessel containing gas. The larger the velocity, the more the momentum transfer, the larger the force is. However, no problem arises in this case since this resistance force is not proportional to the velocity in general, but to the relative velocity with respect to the air; there is a special frame of reference in this case (the air), but not in the case of the magnetic force. The second problem with Newtonian mechanics The second striking difference of the magnetic force is its direction. The Lorentz force is directed perpendicularly to both magnetic field and the velocity of the charge. If the charge is moving in the same direction as the field, there is no magnetic force acting on it. Problems with this facts intersect with problems related to the force depending on the velocity, but the mere fact that no mechanical phenomenon has such features seriously questions the possibility of a mechanistic description of magnetism. Since a moving charge creates a magnetic field and the latter acts on the moving charge, the moving charges may interact magnetically: q1q2 v1v2 F~ 2 2 d c Parallel currents act on each other with magnetic forces – opposite currents repel each other while the currents directed in the same direction attract… I1 I 2 F~ ∆l d – similar to the Coulomb law EXAMPLES: Electric motors Moving charges create a magnetic field. How about moving magnets ? A moving magnet induces an electric field – is a source of an electric current. If a closed loop is moving in the vicinity of a magnet, an electric current is induced in it Generators Regenerative braking Eddy currents APPLICATION: TRANSFORMER How does it work ? Voltage is the same in each loop and adds up: V ~ N Vout N out = Vin N in INPUT: AC current Produces a changing magnetic field Induces a current in the output coil Example: You connect a stereo to the 110 V outlet. The stereo needs 11 V voltage. You have a transformer with 100 turns in the input coil. How many do you need in the output coil ? Vout N out = Vin N in (11 V)/(110 V) = Nout / 100 (11 V)/(110 V) = 1/10 = Nout/100 1/10 x 100 = 10 = Nout Superconductivity In 1911, Kamerlingh Onnes discovered that at very low temperatures, the resistance in metal conductors vanishes Meissner discovered that the superconductors also do not allow the magnetic field to go through them It turned out that the regular resistance does not vanish at all, but the super current is flowing along the surface. This current does not experience resistance and it also compensates the magnetic field inside the superconductor Superconductivity continued The phenomenon of superconductivity was only understood in 1950s Different types of superconductors were discovered including High TC superconductors whose critical temperature can be as high as 150K. Electromagnetism: Electric field Charge Changing MOVING MOVING Magnetic field Magnet ELECTROMAGNETISM 1. A moving charge, an electric current, or changing electric field induce a magnetic field 2. A changing magnetic field induces an electric field. 3. The electric charge is the source of electric and magnetic fields 4. Electric and magnetic fields are related with each other Maxwell equations – complete picture of electro-magnetism Predicted Electro-Magnetic waves Light – EM wave James Clerk Maxwell (1831 – 1879) The new way of description of physical phenomena Maxwell’s description of the electromagnetism is especially important because it is a field description rather than force description. Maxwell’s equations are equations for the electric and magnetic fields that adequately describe the properties of these fields and completely cover all electromagnetic phenomena. They have a strong predictive power. They include possible sources of the fields such as charges or currents, and predict and describe the electromagnetic waves without any charges in the vicinity. Since then, all new phenomena on the fundamental level has been described in the same fashion – in the field language. Electromagnetic Waves Wave: A traveling disturbance consisting of coordinated vibrations that transmits energy but not matter. EM waves: 1. Source of EM waves 2. What oscillates? 3. EM waves propagation Heinrich R. Hertz 1857 – 1894 If the electric field is disturbed by moving charges back and forth… what happens? • The moving charge creates an oscillating electric field and an oscillating magnetic field. • The oscillating magnetic field produces a new changing electric field opposing the original electric field. • The oscillating electric field will produce a new opposing magnetic field. An oscillating (accelerating) charge is a source of EM wave • EM wave is transverse – E and B fields are mutually perpendicular and both perpendicular to the direction of propagation • Both E and B fields oscillate in phase • EM wave propagates with speed of light • does not need a medium… What kind of wave is an electromagnetic wave ? 1. Transverse Wave 2. Longitudinal wave 3. Neither Differences between EM waves and mechanical waves: • EM waves are really two (coupled) waves: an electric field and a magnetic field wave. • EM waves do not require a medium and can travel through vacuum. SPEED of EM waves: In vacuum: c = 299,792,458 m/s c ≈ 300,000 km/s = 186,000 miles/s In a medium they are slower c medium = c/n n – index of refraction A radio station transmits EM waves with frequency 100 MHz. What is the wavelength of the EM waves ? v=c c = fλ c λ= f f = c λ f = 100 MHz = 100 x 106 Hz = 100,000,000 Hz v=c=fλ λ= c/f = (300,000,000 m/s )/(100,000,000 Hz) = 3 m Or alternatively: λ=c/f = 3∗108m/s/108Hz = 3m RADIO WAVES: EM Waves with f = 100 … 109 = 1,000,000,000 Hz λ = 3000 km … 0.3 m Can be generated directly by moving charges back and forth (changing electric field) in an antenna. • Radio: AM = 700 – 1400 kHz, FM = 88 – 108 MHz • TV • Emissions from Planets and stars Very Large Array, New Mexico MICROWAVES: EM waves with f = 109 … 1012 Hz l = 30 cm … 0.3 mm Can be generated by very sophisticated electronics and antennas. • Communication: Satellites • Radar • Cooking food • MRI Radar: Microwaves bounce off metallic objects (Reflection). Measure time it takes to reach object and return. If it takes 20 microseconds for the signal to return, how far away is the airplane ? 20 µs = 20 x 10-6 s = 0.00002 s d = c t/2 = (300,000 km/s)(0.00002 s)/2 = 3 x 105 km/s x 2 x10-5 s /2 = 3 km Venus’ surface viewed by radar: Microwave oven: Water molecules have are electric dipoles. Hydrogen, positive Oxygen, negative Water molecules rotate. When heated, they rotate more vigorously. If we apply a changing electric field to the water, it will be forced to rotate, increasing its kinetic energy. … and its temperature! INFRARED EM waves with f = 1012 … 4 x 1014 Hz λ = 0.3 mm … 0.75 µm (750 nm) Emitted by warm objects, lasers, LEDs • Heat radiation • Remote controls • Some Wireless devices • Lasers • Fiber-optic communication Infrared Photography: VISIBLE LIGHT: EM waves with f = 4 x 1014Hz … 7.5x 1014Hz λ = 750 nm … 400 nm Very narrow range of EM waves which happens to be detectable by human eyes. • Seeing • Optics • TV • Photography • Telescopes • Microscopes Why do we see visible light and not other EM waves ? ULTRAVIOLET EM waves with f = 7.5 x 1014 … 1018 Hz λ = 400 nm … 0.3 nm Emitted by the sun, very hot objects • Tanning • Lithography to make computer chips X RAYS EM waves with f = 1016 … 1020 Hz λ = 30 nm … 0.3 pm Made by bombarding a target with electrons. Can travel through matter almost unhindered. • X-ray imaging • X-ray diffraction Materials with “heavier” atoms in them stop x-rays more efficiently, for example, calcium in bones. Since X-rays have wavelength the size of atoms, they can reveal atomic structure of crystals and molecules. For example: Structure of DNA How are X-rays generated ? Electron gun Electrons X-rays Target GAMMA (γ) RAYS EM waves with f = 3 x 1019 … > 1023 Hz λ = 10 pm … < 3 fm Emitted In nuclear processes, such as γ radioactivity In the process of electron-positron annihilation Bremsstrahlung Supernovae explosions Neutron star collisions, etc. Passage of the EM waves through the atmosphere EM waves passage through the Earth’s atmosphere: Ionosphere: Ions high up (~ 90 km) in atmosphere can reflect certain radio waves (shortwave). Greenhouse effect: Certain gases (H2O, CO2, CH4) reflect or absorb infrared radiation. Keep heat from escaping earth into space. •Keeps it about 35o higher than without… Important for life conditions •Regulates temperature on Earth •Responsible for high temperature on Venus (460 C) – runaway greenhouse effect •Global warming Ozone layer At about 20-40 km above sea level: High concentration of ozone (O3) Stops UV light, protects life on earth. Ozone hole: Certain pollutants can reduce amount of ozone in ozone layer (CFC’s) Do not confuse ozone problem with the greenhouse effect Light and optics Light: • Light is an electromagnetic wave with frequencies in the range of 4 x 1014 to 7.5 x 1014 Hz • In air & vacuum, wavelengths range from 450 nm to 750 nm, where 1 nm = 1 billionth of a meter. • Color is determined by frequency (wavelength) • White light is a mixture of all colors • In vacuum, the speed of light is 3 x 108 m/s. DIFFRACTION: Light passing through a very narrow slit will spread out. Every point reached by the wave (including those in the slit) becomes a source of waves The resultant signal at any point is a result of interference from all directly arriving waves INTERFERENCE: summation of waves – superposition principle Constructive Interference Destructive Interference Destructive: Constructive: Waves in opposite phases Waves in the same phase Interference pattern Slits LAMP Two slit interference Summation of waves with different paths – for a maximum the difference in paths has to be the integer number of wavelengths - for a minimum – half integer Interference effects: thin films Some wavelengths interfere destructively, some constructively Polarization In most light, the E-vector points in random directions, but the direction of the magnetic field is always determined by the direction of the electric field Polarized light, e.g. laser light: Unpolarized light can be polarized by • Reflection • Passing through a polarization filter (polaroid) Note: B field direction and magnitude is always related to the E field’s direction and magnitude Polarized plated filter out different polarizations leaving only one – along its axis Liquid Crystal Displays No signal – transparency because of correct rotation of polarization Electric signal changes polarization in the crystal – no transparency - image The Law of Reflection: The angle of incidence equals the angle of reflection. Reflection: Specular reflection Specular reflection – depends on the reflecting surface only – all rays are reflected similarly – the surface is flat enough - Depends on the wavelength –the shorter the wavelength, the better quality mirror is required for to obtain the specular reflection Diffuse reflection: Reflection off rough surfaces – most common. Most light we see is diffuse reflected light. Colored Objects: • Reflect only some frequencies of light and absorb others. If an object appears red, it Reflects red and absorbs other colors such as blue, yellow, green, etc. REFLECTION, plane mirror When you look into a mirror, what is reversed ? 1. Nothing is reversed. 2. Left and Right are reversed. 3. Up and Down are reversed. 4. Front and Back are reversed. Up Front Right http://perg.phys.ksu.edu/vqm/laserweb/Java/MirrImge/Imageme1.htm One-way mirrors Normally, part of light is reflected and the rest is transmitted Curved mirrors Convex mirror Van Eyck: “The Arnolfini couple” An image from a convex mirror – always virtual, always smaller than the object Concave mirror – virtual image if the object is closer than the focus object Virtual image Enlarged & upright Concave mirror – real image if the object is farther than focus Real image reduced & inverted Plane mirror Concave mirror Convex mirror Curved Mirrors Application: TELESCOPES Aberrations: Spherical aberration Hubble space telescope Tiny error in mirror, repaired in 1993 2.4 m mirror, too flat on one edge by 1 / 50th of the width of a single human hair before after Why ? 1. The pencil actually bends when in contact with water. 2. It’s some kind of interference effect. 3. It’s a magic trick. REFRACTION! When light enters a medium it slows down. Now assume light hits a boundary (= interface) between two media under an angle: What happens ? 1. Some of it reflects off the interface. 2. Some gets transmitted, but how ? Analogy: Car leaving road and entering mud Because the right wheel slows down first, the car rotates. Light does the same thing when it crosses the interface between two different media: = REFRACTION LAW OF REFRACTION: A light ray bends towards the normal when it enters a transparent medium in which light travels slower. It bends away from the normal if it enters a medium in which light travels faster. A 1 2 B Which way is faster for the light, 1 or 2 ? Glass into air Air into glass How to explain the pencil in water ? Image From more optically dense medium to less optically dense medium: What happens if the incident angle is increased? It increases to some maximum angle (critical angle), at which something strange happens: the light does not come out from the more dense medium If this angle is exceeded: it is completely reflected – total internal reflection TOTAL INTERNAL REFLECTION Happens after exceeding the CRITICAL ANGLE Application: Optical fiber LENSES AND IMAGES: Recall: Light rays can be focused by a curved mirror. They can also be focused by using refraction: Convex surface Opposite case: Concave surface Lenses: = Combinations of concave and convex surfaces, utilizing refraction to manipulate light Two convex surfaces = biconvex lens How does it work ? Converging Lens Diverging Lens Biconcave Lens Image formation Lenses are used to form images of objects. How they do that can be determined by “ray tracing”. Optical Axis Focal points Example: In this case light re-converges and projects a real image that is inverted. Another example: What’s different ?? image The light rays are not converging, the lens is not projecting an image. Looking through the lens a virtual image appears. = Magnifier Is the image in a camera real or virtual ? 1. Real 2. Virtual Lens formula sf p= s− f Example: In a slide projector a slide is located 11 cm from a lens with a 10 cm focal length. Where should the screen be located to get a sharp image ? sf p= s− f p = (10 x 11)/(11-10) = 110/1 = 110 cm = 1.1 m Example: A magnifying glass has a focal length of 10 cm. You place a coin at 5 cm from the lens. Where is the image? sf p= s− f p = (5 x 10)/(5-10) = 50/(-5) = -10 cm What does that mean ?? 1. There is no image 2. There is a real image 3. There is a virtual image Magnification: −p M= s Example: Magnifier, s = 5 cm, p = -10 cm M = -(-10)/5 = 10/5 = 2 Virtual upright image Example : slide projector, s = 11 cm, p = 110 cm M = -(110)/11 = -10 What does that mean ? Real, inverted image Telescope, microscope, etc. The object – first lens – first image – second lens – second image and so on. The human eye Focusing Nearsightedness DISPERSION Refraction depends on wavelength of light (color) In glass, shorter wavelengths travel slower than longer wavelengths. Prism Rainbow and halo are results of collective refraction and reflection: Primary rainbow Secondary rainbow Halo effect Blue sky and red sunsets – Scattering of light Where are we? Historically – at the end of the 19th century Mechanics is seemingly complete – Newton’s laws and applications Electricity and magnetism – Maxwell’s equations and applications, waves The end? End of the XIX century – a time of decadence !! What about relation between them – mechanics and E&M? There are two approaches to this question – theoretical and experimental Since they were independent (historically) we will start with the theoretical Principle of relativity – Galileo – Newton: 1. All inertial systems of reference (those that are moving at constant velocities) are equivalent – there is no difference who is moving and who is at rest. Example: if you are driving a car at a constant velocity, you are moving with respect to trees, but they are moving with respect to you. 2. The same laws of physics are valid in any inertial frame of reference. 3. The rules of changing the frame of reference: t = t' time is absolute d = d '−vrel t distance is corrected by the distance passed by the origin v = v'−vrel velocity is corrected by the velocity of the origin Example: You are in a train that is moving at 100 mph. You define your frame – the frame of the train – nonprimed, the frame related to the ground - primed You are walking along the car in the direction of motion of the train at 3 mph v = 100 mph rel t = t' time is absolute d = d '−vrel t distance is corrected by the distance passed by the origin v = v'−vrel velocity is corrected by the velocity of the origin You start your watch at t=0, d=d’=0, then d ' = d + vrel ⋅ t = v ⋅ t + vrel ⋅ t = 3t + 100t = (3 + 100)t = 103t v' = v + vrel = 3 + 100 = 103 Time is absolute, but distance is relative Theoretical problems: If one tries to rewrite beautiful Maxwell equations in a frame moving at a velocity v, i.e. study E&M in a moving frame, using Galilean transformations, the Maxwell equations become very very ugly. This conspicuous ugliness suggests that there is only ONE frame of reference – the absolute frame of reference where they are not ugly. Lorentz force (magnetic force) manifestly dependent on the velocity is another problem. The inertial frames of reference are good for mechanical phenomena, but not for the electromagnetic. Experimental reasons Most physicists of the XIX century agreed that EM waves light included propagate in a special medium called ether. The ether was believed to be massless and not interacting with matter. To verify this hypothesis, the speed of ether was measured in different directions by Michelson and Morley with a surprising but consistently confirmed result – the speed of light is the same in all directions. Another, Trouton – Noble experiment also confirmed that there is no “ether wind”. Dilemma: we live in the “best of the worlds” where the ether is at rest, elsewhere the physics (E&M) is ugly or there is no ether at all and the Galilean transformations are not valid for EM waves. The dilemma The problem can be stated in the following form there are three statements Galilean transformations of coordinates and absolute time. The speed of light is constant in all frames of reference. All laws of Nature are the same in all inertial frames of reference. These three statements contradict each other. One has to be dropped to resolve the problem. The special theory of relativity Einstein’s solution – special theory of relativity: two postulates, 1905: All inertial frames of reference are equally suitable for the description of physical phenomena. The speed of light in vacuum is the same for all observers and is independent of the motion of the source. The toll: Lorentz transformations for time and coordinates replace Galileo transformations vrel t '− 2 d ' c t= time is relative 2 v 1 − rel2 c d '−vrel t ' d= distance measured in a moving frame 2 v 1 − rel2 c v'−vrel v= new formula for velocity addition v' vrel 1− 2 c v + vrel v' = vvrel 1+ 2 c Notice, that Lorentz transformations become Galilean if the ratio v/c is small, v/c << 1, or v << c Consequences: 1. Relativity of simultaneity: If two events occur at different points in the rest frame simultaneously – at the same time, the time interval between these events in the moving frame will not be zero, but proportional to the distance between these points 2. Time dilation: The time interval appears to be longer to the moving observer than it does to the one at rest with respect to the clock. 2. Time dilation continued: prime denotes the frame which is moving with respect to the clock. ∆t ' = ∆t v2 1− 2 c Example: a lifetime of a muon (a particle) is 2.2x10-6s, how long will it live for a stationary observer if the muon moves at 0.8c ? How far will it move before decaying? −6 ∆t = 2.2 ⋅10 2.2 ⋅10 −6 −6 2.2 ⋅10 ∆t ' = = = 3.67 ⋅10 −6 s 2 1 − 0.64 (0.8c) 1− c2 d ' = c ⋅ ∆t ' = 3 ⋅108 ⋅ 3.67 ⋅10 −6 = 1,100 m 3. Lorentz contraction: Found by a moving observer, the length in the direction of motion is contracted: v2 L' = L ⋅ 1 − 2 c v1 + v2 v= v1v2 1+ 2 c 4. Velocity addition formula Example: An observer measures the speed of light coming from the headlights of my car. Extreme case: c+c vres vres v+c v+c v+c = = = =c v c+v vc 1+ 2 1+ c c c 2c 2c c+c = = = =c cc 1 + 1 2 1+ 2 c Dynamical consequences: E0 = mc 2 E= p= Rest energy mc 2 2 v 1− 2 c mv v2 1− 2 c Total enrgy (rest and kinetic) Linear momentum 1. Energy and mass are equivalent 2. Kinetic energy at v << c is equal to mv2/2 3. Energy – momentum conservation in collisions 4. Momentum always conserves! As a result Einstein has pacified Maxwell Electricity and magnetism with Newtonian mechanics. 1. No medium is needed for EM wave propagations 2. EM waves propagate in vacuum at speed of light, same in all directions and frames. 3. EM waves carry both energy and momentum. Experimental evidence: 1. Michelson – Morley experiment, 1887 – speed of light is the same in all directions 2. Trouton – Noble experiment – no ether wind.