David J. Starling Penn State Hazleton PHYS 212

and and The term inductance was coined by Oliver Heaviside in February 1886. David J. Starling Penn State Hazleton PHYS 212

and and Objectives (a) Determine the EMF and electric field induced by a changing magnetic flux and determine the direction of this induced EMF. (b) Calculate and explain the process of self- and mutual-inductance of circuits. Relate mutual inductance to the design of a transformer. (c) Describe the behavior of an RL circuit and determine the characteristic time constant for the system. (d) Calculate the energy in a magnetic field or a magnetic dipole in an external field.

and We have seen electric flux: Φ E = E d A But we can define the magnetic flux in the same way: Φ B = B d A This is the flux through a loop of wire If B is uniform and perpendicular to the loop: Φ B = BA Magnetic flux has units of T m 2, also called the weber (Wb)

Question 1 The ring is rotated clockwise at a constant rate. Which graph best represents Φ B? and

What happens if I increase the flux through some loop? and Current flows in the wire! The faster we change the flux, the bigger the current We induce an emf E in the loop: E = dφ B dt

and What if there are N turns in my loop (solenoid)? Each turn has an induced emf, so we get: E = N dφ B dt We can change Φ B by Increasing/decreasing the magnetic field B Increasing/decreasing the area A Changing the tilt between B and A

and Example 1 A solenoid with 220 turns/cm carries a current of 1.5 A. The current is steadily dropped to 0 over 25 ms. What is the magnitude of the induced emf in the coil C (diameter 2.1 cm and 130 turns) that sits inside the solenoid?

The induced emf in a loop is due to an electric field pushing charges around! Work done: W = qe Also work done: W = F d s = q E d s Therefore: E = E d s So Faraday s Law becomes: E = E d s = dφ B dt and

Example 2 For the copper loop in the figure below, (a) find the magnitude of the induced electric field E at a position r for a uniform, time-dependent magnetic field; (b) find E when r = 5.2 cm and db/dt = 0.13 T/s. and

So what s with the negative sign? The change in flux induces a current The induced current creates a magnetic field This induced magnetic field fights the change in flux There is an opposition to the change E = dφ B dt and

and For an increasing B field For a decreasing B field

Question 2 A coil of wire approaches a long current at a constant speed. and (a) A current is induced in the coil. (b) The rectangle will be pulled in the direction of the current in the wire. (c) A magnetic force acts on the loop that pushes the loop into the page. (d) There is no effect on the coil.

Example 3 A half-circle of radius r = 0.2 m lies in a uniform magnetic field that varies in time according to B = 4t 2 + 2t + 3 (B in T, t in sec), out of the page. The circuit is connected to a 2 V battery and has a total resistance of 2 Ω. and (a) What is the induced emf at t = 10 s? (b) What is the current in the loop at t = 10 s?

Example 4 A rectangular wire loop with height H = 2.0 m and width W = 3.0 m is subjected to a non-uniform magnetic field B = 4t 2 x 2 in S.I. units. Find the induced emf in the loop at time t = 0.10 s. and

When we put a solenoid in a circuit, how does the circuit behave? If we pass a current through a solenoid, it produces a magnetic field, B = µ 0 ni The flux through the solenoid is Φ B = BA If the solenoid has N loops, the total flux through the whole solenoid is NΦ B, called magnetic flux linkage The ratio of this total flux to the current is the inductance L: Compare to C: L = NΦ B i and C = q V

and For an ideal solenoid, L = NΦ B i = (nl)(ba) i = nl(µ 0ni)A i = µ 0 n 2 la The inductance has units of T-m 2 /A, which we call a henry (H). Let s apply Faraday s Law to an inductor: E = d(nφ B) dt = d(li) dt = L di dt Apparently, an inductor produces its own emf This is called self inductance

In a circuit, an inductor acts similarly to a battery, producing an emf E(t) = L di/dt; we include it in the loop rule. and E ir L di dt = 0 di dt + R L i = E L What is i?

Charging up the current: The solution to di dt + R L i = E L and is i(t) = E R ( ) 1 e t/τ L τ L = L/R. Compare to the RC circuit: V = E ( 1 e t/τ )

Dis-charging the current: and The solution to di dt + R L i = 0 is i(t) = E R e t/τ L τ L = L/R. Compare to the RC circuit: V = Ee t/τ

and Key points: The inductor is slow to react Initially, the current is zero Eventually, it provides no resistance This process is exponential with τ = L/R

Example 5 In the circuit below, E = 18 V, R = 9.0 Ω and L = 2.0 mh. (a) Just after the switch is thrown, what is the total current in the circuit? (b) A long time later, what is the total current in the circuit? and

and Example 6 A solenoid with L = 53 mh and R = 0.37 Ω is connected to a battery. How long will it take for the current to reach half its final equilibrium value?

Consider a loop of wire being pushed at a speed v: and An emf is induced in the loop E = dφ B /dt = d(blx)/dt = BLdx/dt = BLv The resulting current is i = E/R = BLv/R The resulting force is F = ilb sin(90 ) = B 2 L 2 v/r Power is just P = Fv, so P = B2 L 2 v 2 R

and In fact, even if it s just a slab of metal, this emf still generates currents: These are called eddy-currents The eddy currents cause a force to oppose the motion

What if we put two coils next to each other? and The magnetic field from coil 1 changes the flux through coil 2 Φ 21 This induces a current in coil 2 This is called mutual inductance: M = N 2Φ 21 i 1 = N 1Φ 12 i 2

and The mutual inductance gives rise to an induced emf: M = N 2Φ 21 i 1 = N 1Φ 12 i 2 E 2 = M di 1 dt E 1 = M di 2 dt

Question 3 In each of the three cases shown, a time-varying current is flowing through the larger coil that produces a magnetic field. Which orientation has the largest mutual inductance? and (a) Case A (b) Case B (c) Case C (d) All the same (e) Not enough information

and Example 7 Two coils are fixed in place. When coil 1 has no current and the current in coil 2 increases at a rate of 15.0 A/s, the emf in coil 1 is 25.0 mv. (a) What is their mutual inductance? (b) When coil 2 has no current and coil 1 has a current of 3.60 A, what is the flux linkage in coil 2?