# Lecture 22. Inductance. Magnetic Field Energy. Outline:

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1 Lecture 22. Inductance. Magnetic Field Energy. Outline: Self-induction and self-inductance. Inductance of a solenoid. The energy of a magnetic field. Alternative definition of inductance. Mutual Inductance. 1

2 - CH2 exam average ~54 %. - To the right are VERY tentative correlations between percentage scores and letter grades in Physics 227. The instructors reserve the right to adjust these boundaries (both up and down) before final grades are assigned at the end of the semester. A B+ < B < C+ < C < D < < F 2

3 Maxwell s Equations Gauss s Law: charges are sources of electric field (and a non-zero net electric field flux through a closed surface); field lines begin and end on charges. ssss E da = Q eeee ε 0 No magnetic monopoles; magnetic field lines form closed loops. ssss B da = 0 Faraday s Law of electromagnetic induction; a time-dependent Φ B generates E. looo E dl = d B da suuu E = dφ B Generalized Ampere s Law; B is produced by both currents and time-dependent Φ E. looo E B dl = μ 0 J + ε 0 suuu da The force on a (moving) charge. F = q E + v B 3

4 Iclicker Question loop 1 loop 2 Two identical concentric loops are arranged as shown in the Figure. One loop has a steady current flowing through it (provided by a power supply). When the power is turned off, in what direction does the induced current flow in loop 2? A. It flows clockwise. B. It flows counterclockwise. C. There is no induced current. D. Something else happens. E. The answer depends on what direction the current was flowing in loop 1. 4

5 Iclicker Question loop 1 loop 2 Two identical concentric loops are arranged as shown in the Figure. One loop has a steady current flowing through it (provided by the power supply). When the power is turned off, in what direction does the induced current flow in loop 2? A. It flows clockwise. B. It flows counterclockwise. C. There is no induced current. D. Something else happens. E. The answer depends on what direction the current was flowing in loop 1. 5

6 Iclicker Question loop 1 loop 2 Two identical concentric loops are arranged as shown in the Figure. One loop has a steady current flowing through it (provided by the power supply). When the power is turned off, will the two loops briefly attract or repel each other? A. They will attract each other. B. They will repel each other. C. There is no induced current. D. They will neither repel nor attract each other. E. The answer depends on the direction of current in loop 1. 6

7 Iclicker Question loop 1 loop 2 Two identical concentric loops are arranged as shown in the Figure. One loop has a steady current flowing through it (provided by the power supply). When the power is turned off, will the two loops briefly attract or repel each other? A. They will attract each other. B. They will repel each other. C. There is no induced current. D. They will neither repel nor attract each other. E. The answer depends on the direction of current in loop 1. 7

8 Induced E.M.F. and its consequences 1. FFFFFFF: E = dφ B 2. I t eeeeeeee B t 3. I t B t 8

9 (Self) Inductance Variable battery I t FFFFFFF: E = dφ B B t The flux depends on the current, so a wire loop with a changing current induces an aitional e.m.f. in itself that opposes the changes in the current and, thus, in the magnetic flux (sometimes called the back e.m.f.). The back e.m.f. : E = dφ B = L L the coefficient of proportionality between Φ and I, purely geometrical quantity. Inductance: (or self-inductance) L Φ I Units: T m 2 /A = Henry (H) The self-inductance L is always positive. This definition applies when there is a single current path (see below for an inductance of a system of distributed currents J r ). Inductance as a circuit element: 9

10 Inductance of a Long Solenoid Long solenoid of radius r and length l with the total number of turns N: L Φ B I l The magnetic flux generated by current I: Φ B = Bπr 2 N = B = μ 0 N l I = μ 0N 2 I l πr 2 = μ 0 n 2 I l πr 2 L = Φ B I = μ 0N 2 l πr 2 = μ 0 n 2 l πr 2 (n the number of turns per unit length) Note that - the inductance scales with the solenoid volume l πr 2 ; - the inductance scales as N 2 : B is proportional to N l and the net flux N. Let s plug some numbers: l =0.1m, N=100, r=0.01m L = μ 0 n 2 l πr 2 = 4π π μμ How significant is the induced e.m.f.? For ~104 A/s (let s say, I~ 1 ma in 0.1 µs) E = L = H A s = 0.4V 10

11 Iclicker Question A current i flows through an inductor L in the direction from point b toward point a. There is zero resistance in the wires of the inductor. If the current is decreasing, A. the potential is greater at point a than at point b. B. the potential is less at point a than at point b. C. the answer depends on the magnitude of di/dt compared to the magnitude of i. D. The answer depends on the value of the inductance L. E. both C. and D. are correct. 11

12 Iclicker Question A current i flows through an inductor L in the direction from point b toward point a. There is zero resistance in the wires of the inductor. If the current is decreasing, A. the potential is greater at point a than at point b. B. the potential is less at point a than at point b. C. the answer depends on the magnitude of di/dt compared to the magnitude of i. D. The answer depends on the value of the inductance L. E. both C. and D. are correct. 12

13 Inductance vs. Resistance Inductor affects the current flow only if there is a change in current (di/dt 0). The sign of di/dt determines the sign of the induced e.m.f. The higher the frequency of current variations, the larger the induced e.m.f. 13

14 Energy of Magnetic Field To increase a current in a solenoid, we have to do some work: we work against the back e.m.f. V i t Let s ramp up the current from 0 to the final value I (i t is the instantaneous current value): V t = L t P t = V t i t = L t i t i t The net work to ramp up the current from 0 to I : = P t f U = P t i P(power) - the rate at which energy is being delivered to the inductor from external sources. I = LL 0 = 1 2 LI2 U = 1 2 LI2 This energy is stored in the magnetic field created by the current. L = μ 0 n 2 l πr 2 = μ 0 n 2 vvvvvv U B = 1 2 μ 0n 2 vvvvvv I 2 = B = μ 0 nn = B2 2μ 0 vvvvvv = u B vvvvvv The energy density in a magnetic field: u B = B2 2μ 0 (compare with u E = This is a general result (not solenoid-specific). ε 0E2 2 ) 14

15 Quench of a Superconducting Solenoid MRI scanner: the magnetic field up to 3T within a volume ~ 1 m 3. The magnetic field energy: U = vvvvvv u B = 1m 3 3T 2 2 4π 10 7 WW A m 4MM The capacity of the LHe dewar: ~2,000 liters. The heat of vaporization of liquid helium: 3 kj/liter. Thus, ~ 1000 liters will be evaporated during the quench. Magnet quench: 15

16 Another (More General) Definition of Inductance Alternative definition of inductance: 1 2 LI2 = B2 2μ 0 dτ all sssss L 2U B I 2 Advantage: it works for a distributed current flow (when it s unclear which loop we need to consider for the flux calculation). 16

17 Example: inductance per unit length for a coaxial cable I I b a Uniform current density in the central conductor of radius a: b U B = 1 2μ 0 B r 0 2 dτ B r = μ 0 I, a < r < b 2ππ μ 0 Ir, r < a 2πa2 at home L = 2U B I 2 Straight Wires: Do They Have Inductance? Using L = 2U B /I 2 : U B = 1 2μ 0 a b μ 0I 2ππ 2 Using L = Φ/I: ~l B μ 0I 2ππ l dτ b = 1 2μ 0 a μ 0I 2ππ l Φ l μ 0I 2ππ a 2 l 2ππππ = μ 0I 2π l ll l a (b~l ) μ 0I 2 4π l ll l a L H μ 0 2π l ll l a l m ll l a l = 1cc L 10 9 H L H μ 0 2π l ll l a 17

18 Inductance vs. Capacitance L = Φ B I = 2U B I 2 C = Q V = 2U E V 2 U B = 1 2 LI2 = 1 2 Φ B 2 L U B = B2 2μ 0 vvvvvv = u B vvvvvv Q 2 U E = 1 2 CV2 = 1 2 C U E = ε 0E 2 2 vvvvvv = u E vvvvvv 18

19 Mutual Inductance Let s consider two wire loops at rest. A time-dependent current in loop 1 produces a time-dependent magnetic field B 1. The magnetic flux is linked to loop 1 as well as loop 2. Faraday s law: the time dependent flux of B 1 induces e.m.f. in both loops. B 1 I 1 loop 2 loop 1 Primitive transformer The e.m.f. in loop 2 due to the time-dependent I 1 in loop 1: E 2 = dφ 1 2 Φ 1 2 = M 1 2 I 1 The flux of B 1 in loop 2 is proportional to the current I 1 in loop 1. The coefficient of proportionality M 1 2 (the so-called mutual inductance) is, similar to L, a purely geometrical quantity; its calculation requires, in general, complicated integration. Also, it s possible to show that M 1 2 = M 2 1 = M so there is just one mutual inductance M. M can be either positive or negative, depending on the choices made for the senses of transversal about loops 1 and 2. Units of the (mutual) inductance Henry (H). E 2 = M I 1 M = Φ 1 2 I 1 19

20 Calculation of Mutual Inductance Calculate M for a pair of coaxial solenoids (r 1 = r 2 = r, l 1 = l 2 = l, n 1 and n 2 - the densities of turns). l B 1 = μ 0 n 1 I 1 Φ = B 1 πr 2 = μ 0 n 1 I 1 πr 2 M 1 2 = M 2 1 = M L = μ 0 n 2 l πr 2 = μ 0N 2 πr 2 Experiment with two coaxial coils. l the flux per one turn M = N 2Φ = N I 2 μ 0 n 1 πr 2 = μ 0N 1 N 2 πr 2 1 l M = L 1 L 2 The induced e.m.f. in the secondary coil is much greater if we use a ferromagnetic core: for a given current in the first coil the magnetic flux (and, thus the mutual inductance) will be ~K m times greater. E 2 = M di 1 = K m M di 1 no-core mutual inductance 20

21 Next time: Lecture 23. RL and LC circuits

22 Appendix I: Inductance of a Circular Wire Loop 2a b Example: estimate the inductance of a circular wire loop with the radius of the loop, b, being much greater than the wire radius a. The field dies off rapidly (~1/r) with distance from the wire, so it won t be a big mistake to approximate the loop as a straight wire of length 2πb, and integrate the flux through a strip of the width b: b B μ 0I 2ππ 2πb Note that we cannot assume that the wire radius is zero: b Φ 2ππ μ 0I 2ππ = μ 0 bi ll b a a L b = 1m, a = 10 3 m 4π 10 7 L = Φ I μ 0b ll b a WW A m 1m ll103 10μμ b Φ~ μ 0I 2ππ "0" ~ll b "0" - diverges! What to do: introduce some reasonably small non-zero radius. 22

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