Microsoft PowerPoint Chapter 29 Electromagnetic induction 1292021 One Love One Future Chapter 29 ELECTROMAGNETIC INDUCTION Exercises 1, 3, 5, 9, 17, 19, 23, 25, 27, 29, 35, 37, 39 Problems 43(44), 4. Dang Duc Vuong
12/9/2021 Magnetic flux: Faraday’s Law of Induction: The induced emf in a closed loop equals the negative of the time rate of change of the magnetic flux through the loop Chapter 29: ELECTROMAGNETIC INDUCTION Direction : curl fingers of right hand around area vector A Exercises: 1, 3, 5, 9, 17, 19, 23, 25, 27, 29, 35, 37, 39 Problems: 43(44), 45, 49(49), 55(55), 61(61), 63(63), 65(65), 67(67), 69(70), 71(72) Dang Duc Vuong Email: vuong.dangduc@hust.edu.vn N = number of turns Coil: One Love One Future 12/9/2021 Dang Duc Vuong - SEP - HUST One Love One Future 1 12/9/2021 Dang Duc Vuong - SEP - HUST 2 Lenz’s Law Motional Electromotive Force If a conductor moves in a magnetic field, a motional emf is induced The direction of any magnetic induction effect is such as to oppose the cause of the effect - If the flux in a stationary circuit changes, the induced current sets up a magnetic field opposite to the original field if original B increases, but in the same direction as original B if B decreases - The induced current opposes the change in the flux through a circuit Magnetic force FB that causes free charges in rod to move, creating excess charges at opposite ends The excess charges generate an electric field (from a to b) and electric force (F = q E) opposite to magnetic force Induced Electric Fields An induced emf occurs when there is a changing magnetic flux through a stationary conductor Charges are in equilibrium A current (I) in solenoid sets up B along its axis, the magnetic flux is: General form Induced current in loop (I’): I’= ε / R Closed conducting loop The force that makes the charges move around the loop is not a magnetic force There is an induced electric field in the conductor caused by a changing magnetic flux Induced current: a time-varying B induces E in stationary conductor and emf E is induced even when there is no conductor One Love One Future 12/9/2021 Dang Duc Vuong - SEP - HUST One Love One Future 12/9/2021 Dang Duc Vuong - SEP - HUST 12/9/2021 29.43(44) A Changing Magnetic Field You are testing a new data-acquisition system This system allows you to record a graph of the current in a circuit as a function of time As part of the test, you are using a circuit made up of a 4.00-cm-radius, 500-turn coil of copper wire connected in series to a 600- resistor Copper has resistivity 1.72xl0-8 .m, and the wire used for the coil has diameter 0.0300 mm You place the coil on a table that is tilted 30.0° from the horizontal and that lies between the poles of an electromagnet The electromagnet generates a vertically upward magnetic field that is zero for t < 0, equal to (0.120 T)(l - cost) for < t < 1.00 s, and equal to 0.240 T for t > 1.00 s (a) Draw the graph that should be produced by your data-acquisition system (This is a full-featured system, so the graph will include labels and numerical values on its axes.) (b) If you were looking vertically downward at the coil, would the current be flowing clockwise or counterclockwise? Displacement Current and Maxwell’s Equations Conduction current : iC into one plate and out of the other Displacement current (iD): fictitious current in region between capacitor’s plates Generalized Ampere’s Law: Displacement current creates B between plates of capacitor while it charges Use Faraday’s law to calculate the magnitude of the induced emf and Lenz’s law to determine its direction; Ohm’s law to calculate I Maxwell’s Equations of Electromagnetism d d dB NBA cos N r cos30 dt dt dt For t < and t > 1.00 s, dB/dt = = I = /R =0 Ampere’s law B 600 For t 1.00 s, dB/dt = (0.120 T) πsin (πt) N r 0.120 sin t i R i 0.224sin t mA N2r B L R 600 600 the current is clockwise A r Faraday’s law One Love One Future Dang Duc Vuong - SEP - HUST 12/9/2021 Dang Duc Vuong - SEP - HUST 12/9/2021 6 29.49(49) In Fig the loop is being pulled to the right at constant speed v A constant current I flows in the long wire, in the direction shown (a) Calculate the magnitude of the net emf induced in the loop Do this two ways: (i) by using Faraday’s law of induction and (ii) by looking at the emf induced in each segment of the loop due to its motion (b) Find the direction (clockwise or counterclockwise) of the current induced in the loop Do this two ways: (i) using Lenz’s law and (ii) using the magnetic force on charges in the loop (c) Check your answer for the emf in part (a) in the following special cases to see whether it is physically reasonable: (i) The loop is stationary; (ii) the loop is very thin, so a 0; (iii) the loop gets very far from the wire 29.45 A circular coil of wire had a radius of 0.500 m, 20 turns, and a total resistance of 1.57 The coil lies in the xy- plane The coil is in a uniform magnetic field B that is in the -z-direction, which is directed away from you as you view the coil The magnitude B of the field depends on time as follows: it increases at a constant rate from at t = to 0.800 T at t = 0.500 s; is constant at 0.800 T from t = 0.500 s to t = 1.00 s; decreases at a constant rate from 0.800 T at t = 1.00 s to at t = 2.00 s (a) Graph B versus t for t from to 2.00 s (b) Graph the current I induced in the coil versus t for t from to 2.00 s Let counterclockwise currents be positive and clockwise currents be negative (c) What is the maximum induced electric field magnitude in coil during 0- to 2.00-s time interval? d d NBA cos N r dt dt B N R 1.57 i r R R dB dt (a) Consider a narrow strip of width dx and a distance x from the long wire The magnetic field of the wire at the strip is dB dt The flux through the strip is One Love One Future B One Love One Future 12/9/2021 Dang Duc Vuong - SEP - HUST One Love One Future 12/9/2021 d Iabv dt 2r(r a) B Dang Duc Vuong - SEP - HUST 12/9/2021 For a bar of length L moving at speed v perpendicular to a magnetic field B: = BvL I I 0 bv 2 bv 2 r a r Both emfs 1and 3 are directed toward the top of the loop so oppose each other The net emf is (c) Check your answer for the emf in part (a) 0 (i) The loop is stationary v = the induced emf should be zero Ibv Iabv 2 r r a 2r(r a) (b) Find the direction (ii) the loop is very thin, so a 0 When a→ the flux goes to zero and the emf should approach zero (iii) the loop gets very far from the wire (i) Lenz’s law: The flux of the induced current opposes the change in flux the current is clockwise When r →∞ the magnetic field through the loop goes to zero and the emf should go to zero (ii) using the magnetic force: right hand ruler segment B1 B1 > B3 FB1 > FB3 One Love One Future segment B3 the induced current in the loop is clockwise (agree with Lenz’s law) Dang Duc Vuong - SEP - HUST 12/9/2021 9 One Love One Future 12/9/2021 Dang Duc Vuong - SEP - HUST 10 10 29.61(61) The long, straight wire shown in Fig below carries constant current I A metal bar with length L is moving at constant velocity v, as shown in the figure Point a is a distance d from the wire (a) Calculate the emf induced in the bar (b) Which point, a or b, is at higher potential? (c) If the bar is replaced by a rectangular wire loop of resistance R, what is the magnitude of the current induced in the loop? 29.55(55) Terminal Speed A conducting rod with length L, mass m, and resistance R moves without friction on metal rails as shown in Fig below A uniform magnetic field B is directed into the plane of the figure The rod starts from rest and is acted on by a constant force F directed to the right The rails are infinitely long and have negligible resistance (a) Graph the speed of the rod as a function of time (b) Find an expression for the terminal speed (the speed when the acceleration of the rod is zero) B The magnetic field of the wire is given by F and varies along the length of the bar Divide the bar up into thin slices Vba is negative, point a is at higher potential than point b As the loop moves to the right the magnetic flux through it doesn’t change The terminal speed vt occurs when the pulling force is equaled by the magnetic force One Love One Future 11 12/9/2021 Dang Duc Vuong - SEP - HUST 11 One Love One Future 12 12/9/2021 Dang Duc Vuong - SEP - HUST 12 12/9/2021 29.63(63) A slender rod, 0.240 m long, rotates with an angular speed of 8.80 rad/s about an axis through one end and perpendicular to the rod The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of 0.650 T (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at 8.80 rad/s about an axis through its center and perpendicular to the rod In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end? The emf induced in a thin slice is The emf between the center of the rod and each end is the direction of the emf: from the center of the rod toward each end The emfs in each half of the rod thus oppose each other: there is no net emf between the ends of the rod Other method d One Love One Future L d 2 t d dt d BdA B d B L d dt BL 0.165 V dt 2 12/9/2021 Dang Duc Vuong - SEP - HUST 13 13 One Love One Future 12/9/2021 Dang Duc Vuong - SEP - HUST 14 14 29.65(65) A rectangular loop with width L and a slide wire with mass m are as shown in Fig below A uniform magnetic field B is directed perpendicular to the plane of the loop into the plane of the figure The slide wire is given an initial speed of v0 and then released There is no friction between the slide wire and the loop, and the resistance of the loop is negligible in comparison to the resistance R of the slide wire (a) Obtain an expression for F, the magnitude of the force exerted on the wire while it is moving at speed v (b) Show that the distance x that the wire moves before coming to rest is x = mv0R/a2B2 29.67(67) The magnetic field B, at all points within a circular region of radius R, is uniform in space and directed into the plane of the page as shown in Fig (The region could be a cross section inside the windings of a long, straight solenoid.) If the magnetic field is increasing at a rate dB/dt, what are the magnitude and direction of the force on a stationary positive point charge q located at points a, b, and c? (Point a is a distance r above the center of the region, point b is a distance r to the right of the center, and point c is at the center of the region the induced electric field E (a) Use Faraday’s law to calculate the induced emf, Ohm’s law to calculate I Lenz’s law F The induced emf: = Bva the induced current B inside; dl is clockwise E is tangent to the circle in the counterclockwise direction Lenz’s law: (b) Show that the distance x Take +x to be toward the right and let the origin be at the location of the wire at t = 0, so x = One Love One Future 15 12/9/2021 Dang Duc Vuong - SEP - HUST 15 One Love One Future 16 12/9/2021 Dang Duc Vuong - SEP - HUST 16 12/9/2021 29.71(72) A capacitor has two parallel plates with area A separated by a distance d The space between plates is filled with a material having dielectric constant K The material is not a perfect insulator but has resistivity The capacitor is initially charged with charge of magnitude Q0 on each plate that gradually discharges by conduction through the dielectric (a) Calculate the conduction current density jC(t) in the dielectric (b) Show that at any instant the displacement current density in the dielectric is equal in magnitude to the conduction current density but opposite in direction, so the total current density is zero at every instant 29.69(70) Falling Square Loop A vertically oriented, square loop of copper wire falls from a region where the field B is horizontal, uniform, and perpendicular to the plane of the loop, into a region where the field is zero The loop is released from rest and initially is entirely within the magnetic-field region Let the side length of the loop be s and let the diameter of the wire be d The resistivity of copper is R and the density of copper is m If the loop reaches its terminal speed while its upper segment is still in the magnetic-field region, find an expression for the terminal speed At the terminal speed, the upward force FB exerted on the loop due to the induced current equals the downward force of gravity: FB = mg mg FB + i in the direction from the + to - the plate of the capacitor mg The conduction current flows from the positive to the negative plate of the capacitor One Love One Future 12/9/2021 Dang Duc Vuong - SEP - HUST 17 17 One Love One Future 12/9/2021 Dang Duc Vuong - SEP - HUST 18 18 www.hust.edu.vn Thank you for your attentions! One Love One Future 19 12/9/2021 Dang Duc Vuong - SEP - HUST 12/9/2021 19 20 Dang Duc Vuong - SEP - HUST 20 ...12/9/2021 29. 43(44) A Changing Magnetic Field You are testing a new data-acquisition system This system allows... One Love One Future Dang Duc Vuong - SEP - HUST 12/9/2021 Dang Duc Vuong - SEP - HUST 12/9/2021 6 29. 49(49) In Fig the loop is being pulled to the right at constant speed v A constant current I... magnitude of the net emf induced in the loop Do this two ways: (i) by using Faraday’s law of induction and (ii) by looking at the emf induced in each segment of the loop due to its motion (b)