6/3/2013 Ch 34: Faraday’s Experiment Electromagnetic Induction • Trying to induce a current using magnetic fields • No induced current in “Y” loop with a DC circuit • Saw a current when opening and closing the switch (changing the magnetic field) Faraday’s Law - An induced emf is produced by a changing magnetic field – Can move magnet or loop – Direction of motion controls direction of current – No movement, no current Predict the direction of the induced current Motional EMF • A current is caused by an electric field • Current continues until FB = FE FE = qE FB = qvB qE = qvB E = vB 1 6/3/2013 Which is the correct picture? E = Blv (assumes B ┴ to v) An airplane travels at 1000 km/hr in a region where the earth’s magnetic field is 5 X 10-5T (vertical). Calculate the potential difference between the wing tips if they are 70 m apart. 1000 km/hr = 280 m/s E = Blv E = (5 X 10-5T )(70 m)(280 m/s) = 1.0 V Motional EMF • Moving a bar or wire produces charge separation • If looped, produces a current • Bar doesn’t want to move (Lenz’s law), must exert a force • Remember Fmag = IlB Fpull = vl2B2 R l = length R = resistance A metal bar of length l rotates in a magnetic field B that is perpendicular to the plane of rotation. It rotates at an angular speed of w. a. Determine the potential difference between the two ends of the bar (HINT: substitute for w, and then integrate from 0 to l, the pivot to the end) Example Consider the following set-up. The bar is 10.0 cm long. a. Calculate the current needed for the bulb (P = IV) b. Calculate the resistance of the bulb c. Calculate the speed needed to achieve this current. (E = Blv) d. Calculate the force required for the pull 2 6/3/2013 EMF in a Moving Conductor • Slide a conducting bar on the wire loop • Increasing area • What direction is the induced current? (right hand rule) Moving Conductor: Ex 2 Blood contains charged ions. A blood vessel is 2.0 mm in diameter, the magnetic field is 0.080 T, and the blood meter registers a voltage of 0.10 mV. What is the flow velocity of the blood? E = Blv v = E /Bl v = (1.0 X 10-4 V) (0.080 T)(0.0020m) v = 0.63 m/s Magnetic Flux (flow) FB = Magnetic Flux FB = BAcosq B = Magnetic Field (T) A = area passes through (m2) q = Angle ┴ to surface 3 6/3/2013 Lenz’s Law If B ┴ to surface – Cos 0o = 1 – Maximum flux An induced current’s magnetic field opposes the original change in flux If B || to surface • Always tries to keep magnetic field inside loop constant. • Use right-hand rule to predict direction of current. – Cos 90o = 0 – No flux – Curve your fingers around the loop – v is direction of the induced current Lenz’s Law: Ex 1 Why is the direction of the induced current? Lenz’s Law: Ex 2 What will happen to the current if you allow the ring to relax to its original shape? • Larger area • Induced I will reverse direction 3 Ways to cause an emf 1. Change the magnetic field 2. Change area of loop 3. Rotate the loop (or magnet) No flux Maximum flux 4 6/3/2013 Lenz’s Law: Ex 3a Predict the direction of the induced current in the following situations • Counterclockwise current • Magnet is going in (north in), need a current pointing north out through the loop • No current • Magnetic flux is || to the loop • Magnetic field decreasing • Counterclockwise current to increase it 5 6/3/2013 • Decreasing flux • Clockwise current induced • Initially no flux • Flux increases to left • Counterclockwise current B A long straight wire carries a current I as shown. a. Predict the direction of the magnetic field inside the adjacent loop. b. As the wire is pulled away from the loop, predict the direction of the induced current. Faraday’s Law of Induction E = N DFB Dt N = number of loops in a wire DFB/Dt = change in magnetic flux over time 6 6/3/2013 Motional EMF E = DFB Dt E = BDA Dt E = BlvDt Dt Lenz’s Law: Ex 4 A square coil of 100 loops is quickly pulled from the magnetic field as shown in 0.10 s. Calculate the change in flux. E = Blv (assumes B ┴ to v) FBfinal =0 FBinitial = BAcos0 FBinitial = (0.60 T)(0.050m)2(1) FBinitial = 0.0015 Wb DF = FBfinal – Fbinitial DF = 0 – 0.0015 Wb = -0.0015 Wb What Voltage and current are produced in the loop (assume resistance = 100 W) E = -NDFB Dt E = -(100)(-0.0015 Wb) = 1.5 V 0.10 s V = IR I = V/R = 1.5 V/100 W = 0.015 A (15 mA) Faraday’s Law: Ex 1 Induced Electric Fields A patient neglects to remove a 6.0 cm copper bracelet (R = 0.010 W) before getting an MRI. The magnetic field changes from 1.00 T to 0.40 T in 1.2 s. Assume the field passes perpendicular to the bracelet. a. Calculate the magnetic flux for both T’s (FB = BAcosq) b. Calculate the voltage through the bracelet based on the change in flux. c. Calculate the current through the bracelet Coulomb vs. non-Coulomb 1. Coulomb Electric field – created by positive and negative charges 2. Non-Coulomb – created by a changing magnetic field Both exert forces on charges (F = qE) 7 6/3/2013 Another version of Faraday’s Law A 4.0 cm diameter solenoid is wound with 2000 turns/meter. The current oscillates at 60 Hz and has an amplitude (maximum) of 2.0 A. Here are some equations to help you: B = m0nI I = I0sinwt a. Determine formula for the electric field inside the solenoid. b. Determine the maximum electric field inside the solenoid. • Usually want to avoid inductance – Resistors are wound in two directions to cancel the inductance • Acts as a resistor for alternating current (impedance) • Ex Electric Field inside a solenoid Self-Inductance: Solenoids • Solenoid (inductor) – coil of wire (choke coil) • L = inductance of the coil (Henry’s) • As current increases in an inductor, an induced emf is created • Induced emf retards the increase of current (a back emf) Inductance of a solenoid L = m0N2A l m0 = 4p X 10-7 T m/A N = number of turns A = cross-sectional area l = length of solenoid – dc current can burn out a transformer – ac has self-inductance (impedance) that limits the current 8 6/3/2013 Inductance: Ex 1 Calculate the inductance of a solenoid with 100 turns, a length of 5.0 cm, and a cross sectional area of 0.30 cm2. L = m0 Inductance: Ex 2 The same solenoid is now filled with an iron cores (m = 4000). Calculate the inductance L = (4000)(7.5 X 10-6H) L = 0.030 H or 30 mH N2A l L = (4p X 10-7 T m/A)(100)2(3 X 10-5m2) (0.05 m) L = 7.5 X 10-6 H or 7.5 mH Inductance: Ex 3 An inductor is made by tightly wrapping 0.300 mm diameter wire around a 4.00 mm diameter cylinder. Calculate the length of cylinder needed to produce an inductance of 10 mH. (5.7 cm) A 1.0 A current passes through a 10 mH inductor coil. Calculate the potential difference if the current drops to zero in 5.0 ms. Inductors in Circuits Used to store charge at high voltage (spark plug within a car) Energy in Inductors (Energy stored in a solenoid) 9 6/3/2013 A 10 mH inductor is 5.7 cm long and 4.0 mm in diameter. It carries a 100 mA current. a. Calculate the energy stored in the inductor. (5.0 X 10-8 J) b. Calculate the solenoid volume. (7.16 X 10-7 m3) c. Calculate the magnetic field density (U/volume = 0.070 J/m3) d. Calculate the magnetic field strength. (4.2 X 10-4 T) LC Circuits • • • • Oscillating circuit Responds at natural frequency (resonance) Used in cell phones to pick up a signal Dials change in resonance frequency w= 1 LC LC: Ex 1 A 1.00 mH inductor is to be used for an AM radio. Calculate the capacitance needed to pick up a frequency of 902 kHz to listen to Rush Limbaugh. LR Circuits • • • • Electromagnets Radio tuners L is inductance R is resistance of inductor and any other resistance • Inductor smoothes out the voltage drop/increase • Initially very low impedance • Impedance rises with current I = Ioe-t/t t = time constant (time to read 63% of max) t=L R 10 6/3/2013 Turning on current LR Circuits: Ex 1 – Current rises quickly, then levels off A solenoid has an inductance of 87.5 mH and a resistance of 0.250 W. Find the time constant. a. Calculate the time constant b. Calculate the time needed to reach ½ of the maximum current. (I = Ioe-t/t) t = L/R t = 87.5 X 10-3 H/0.250 W = 0.350 s Turning off current – Opposite shape – I = Imax e-t/t Rank in order, from smallest to largest, the time constants of the following circuits. a b The switch in the following circuit has been in position a for a long time (VL = 0). a. Calculate the current initially (100 mA) b. Calculate the current at 5.0 ms after the switch is thrown. (61 mA) c. At what time has the current decayed to 1% of its initial value? (46 ms) c DC vs AC • DC = Direct current – Electrons flow constantly – Electrons only flow in one direction (negative to positive) – Batteries provide DC current DC vs AC • AC = Alternating current – – – – Electrons switch directions “Pulsed current” Home electricity More efficient for power transmission over large distances – USA uses 60 Hertz (60 cycles per second), many other countries use 50 Hz 11 6/3/2013 DC vs AC DC •Electrons flow constantly •Electrons flow in only one direction •Batteries AC •Electrons flow in short burst •Electrons switch directions (60 times a second) •House current 2. 0.10 T 4. 0.040 Wb 6. Bab 8. a) right to left 10. 8.7 X 10-4 Wb 12. 5.0 A 14. 4.7 T/s 16. 0.040 V/m 18. 100 V 20. 0.20 J 22. 0.25 mH b) no induced current 24. 50 mA 26. 6.9 X 10-5 s 28.a) parabola b) 1.6 X 10-3(1 + t) -2 c) 1.7 X 10 A 30. a) Hill b) (0.126)(1-1/2t) c) -63 mA 32. 26 mV 48. a) 4.0 V b) 100 A c) 3.0 V 50. a) 6.3 X 10-4 N b) 48. a) 4.0 V b) 100 A c) 3.0 V 76. a) 76 mA b) 0.50 ms (1st maximum) 12 6/3/2013 Counter EMF Counter EMF: Ex 1 • Counter (back) emf – as a motor turns, an emf is induced that opposes the motion (Lenz’s law) • Counter emf is less than the external voltage when under a load • The slower a motor rotates, the less counter emf The windings of a DC motor have a resistance of 5.0 W. When the motor reaches full speed, the counter emf is 108 V. What is the current when the motor is just starting up, and when it reaches full speed (voltage = 120 V) 13 6/3/2013 Just starting up (almost no counter emf) V = IR I = V/R = 120 V/5.0 W = 24 A At full speed (V = 120 V – 108 V = 12 V) I = V/R = 12 V/5.0 W = 2.4 A Counter EMF: Ex 2 If a blender or drill jams (motor can’t turn), the device may burn out. Why? – No counter emf – Current can be very high – Wires may heat up Current is VERY high at start – Lights may dim when refrigerator starts – Lights dim if on when starting a car Electric Generators (Dynamo) • Generator is the inverse of a motor • AC Generator shown • Rotation through magnetic field induces I • Current flows first one way, then the other • Segments ab and cd are moving conductor • (Side segments have force in wrong direction) E = Blv┴ v┴ = vsinq E = 2NBlvsinq • Can consider angular rotation q = wt v = wr = w(h/2) h = length of ad or bc E = 2NBlvsinq E = 2NBlw(h/2) sin wtlh = Area E = wNBAsin wt Remember w = 2pf f = frequency (Hertz) w (radians/s) 14 6/3/2013 Generator: Ex 1 • Over 99% of electricity in US produced by generators – – – – Coal/oil/gas plants Wind power Nuclear Water • 60 Hz in US and Canada • 50 Hz in some others Generator: Ex 2 A 60-Hz generator rotates in a 0.010 T magnetic field. If the area of the coil is 2.0 m2, how many loops must it contain for a peak output of 160 V? (21 turns) A 60-Hz generator rotates in a 0.15 T magnetic field. If the area of the coil is 0.020 m2, how many loops must it contain for a peak output of 170 V? E = NBAwsin wtassume wt = 90 E = NBAw = E /BAw w = 2pf = 2p(60Hz) = 377 s-1 N = 150 loops DC Generator • Split ring commutator • Many windings smooth out the current Alternator • Engine turns the rotor • Magnetic field produced • Current induce in stationary stator coils 15 6/3/2013 Microphones • Coil moves in and out of magnetic field with sound • emf induced in the coil • Current is then sent to speakers, recorders, etc.. Seismograph • Magnet moves and creates current in coil • Current translated into signal for eart’s movement • Two coils linked by soft iron core • Can be intertwined • Flux from primary induces a current in the secondary (99% efficient) • Vary number of loops to control voltage VS = NS VP NP Tape Heads Recording • Changing current in coil creates magnetic field • Magnetizes the metal on the tape Playback • Changing magnetic field from tape induces current in coil • Digital tape only has 1’s and 0’s Transformers • Increase or decrease AC voltage • TV – increase voltage for picture tube • Power packs – decrease voltage • Utility poles – decrease voltage for house YEAH!!! MY FAVORITE TOPIC!!!! • Step-up Transformer – Increases voltage • Step-down Transformer – Decreases voltage • POWER can’t increase (can’t get something for nothing) P = VI PP = P S VPIP = VSIs IS = NP IP NS 16 6/3/2013 Transformers: Ex 1 A transformer for a radio reduces the voltage from 120 V to 9.0 V. The secondary has 30 turns and the radio draws 400 mA. Calculate the turns in the primary. VS = NS VP NP NP = NSVP = (30)(120V) = 400 turns VS 9V Calculate the current in the primary Calculate the power transformed IS = NP IP NS IP = ISNS = (0.400A)(30) = 0.030 A (30 mA) NP (400) P = IV P = (0.030 A)(120 V) = 3.6 W Transformers: Ex 2 An average of 120 kW of power is sent to a small town 10 km from the power plant. The transmission lines for a total resistance of 0.40 W. Calculate the power lost to resistance if the power is transmitted at 240 V vs. 24,000 V. At 240 V P = IV I = P/V = 120,000 W/240 V = 500 A (can use either primary or secondary) Ploss = I2R Ploss = (500 A)2 (0.40 W) = 100 kW At 24,000 V P = IV I = P/V = 120,000 W/24,000 V = 5 A Ploss = I2R Ploss = (5 A)2 (0.40 W) = 10 W 17 6/3/2013 • Transformers only work on ac • dc only produces a secondary voltage when switch is opened or closed 18