1. Induction and Inductance III
Physics 2113
Jonathan Dowling
Lecture 31: FRI 06 NOV
Induction and Inductance III
Fender Stratocaster
Solenoid Pickup

2. Changing B-Field Produces E-Field!
We saw that a time varying magnetic FLUX creates an induced EMF in a wire, exhibited as a current.
Recall that a current flows in a conductor because of electric field.
Hence, a time varying magnetic flux must induce an ELECTRIC FIELD!
A Changing B-Field Produces an E-Field in Empty Space! B dA
To decide direction of E-field use Lenz’s law as in current loop.

3. Solenoid Example
A long solenoid has a circular cross-section of radius R.
The magnetic field B through the solenoid is increasing at a steady rate dB/dt.
Compute the variation of the electric field as a function of the distance r from the axis of the solenoid. B
First, let’s look at r < R:
Next, let’s look at r > R:
magnetic field lines
electric field lines

4. Solenoid Example Cont. B E E(r) i r r = R
Added Complication: Changing B Field Is Produced by Changing Current i in the Loops of Solenoid! i B E r
E(r)
r = R

5. Summary Two versions of Faradays’ law:
A time varying magnetic flux produces an EMF:
A time varying magnetic flux produces an electric field:

6. 30.7: Inductors and Inductance:
An inductor (symbol ) can be used to produce a desired magnetic field.
If we establish a current i in the windings (turns) of the solenoid which can be treated as our inductor, the current produces a magnetic flux FB through the central region of the inductor.
The inductance of the inductor is then
The SI unit of inductance is the tesla–square meter per ampere (T m2/A). We call this the henry (H), after American physicist Joseph Henry,

7. Inductors: Solenoids
Inductors are with respect to the magnetic field what capacitors are with respect to the electric field. They “pack a lot of field in a small region”. Also, the higher the current, the higher the magnetic field they produce.
Capacitance C how much potential for a given charge: Q=CV
Inductance L how much magnetic flux for a given current: Φ=Li
Using Faraday’s law:
Joseph Henry
( )

9. loop 1
loop 2
(30–17)

11. Example (a) 50 V (b) 50 V i Magnitude = (10 H)(5 A/s) = 50 V
The current in a L=10H inductor is decreasing at a steady rate of i=5A/s.
If the current is as shown at some instant in time, what is the magnitude and direction of the induced EMF?
Magnitude = (10 H)(5 A/s) = 50 V
Current is decreasing
Induced EMF must be in a direction that OPPOSES this change.
So, induced EMF must be in same direction as current
(a) 50 V
(b) 50 V

12. The RL circuit Key insights:
Set up a single loop series circuit with a battery, a resistor, a solenoid and a switch.
Describe what happens when the switch is closed.
Key processes to understand:
What happens JUST AFTER the switch is closed?
What happens a LONG TIME after switch has been closed?
What happens in between?
Key insights:
You cannot change the CURRENT in an inductor instantaneously!
If you wait long enough, the current in an RL circuit stops changing!
At t = 0, a capacitor acts like a solid wire and inductor acts like break in the wire.
At t = ∞ a capacitor acts like a break in the wire and inductor acts like a solid wire.

13. 30.8: Self-Induction:

14. 30.9: RL Circuits:

15. 30.9: RL Circuits:
If we suddenly remove the emf from this same circuit, the flux does not immediately fall to zero but approaches zero in an exponential fashion:

16. 30.9: RL Circuits:

17. RC vs RL Circuits
In an RL circuit, while fluxing up (rising current), E = Ldi/dt and the loop rule mean:
magnetic field increases from 0 to B
current increases from 0 to E/R
voltage across inductor decreases from -E to 0
In an RC circuit, while charging,
Q = CV and the loop rule mean:
charge increases from 0 to CE
current decreases from E/R to 0
voltage across capacitor increases from 0 to E

18. ICPP
Immediately after the switch is closed, what is the potential difference across the inductor?
(a) 0 V
(b) 9 V
(c) 0.9 V
10 Ω
10 H
9 V
Immediately after the switch, current in circuit = 0.
So, potential difference across the resistor = 0!
So, the potential difference across the inductor = E = 9 V!

19. ICPP
40 Ω
10 H
3 V
10 Ω
Immediately after the switch is closed, what is the current i through the 10 Ω resistor?
(a) A
(b) 0.3 A
(c) 0
Immediately after switch is closed, current through inductor = 0. Why???
Hence, current through battery and through 10 Ω resistor is i = (3 V)/(10 Ω) = 0.3 A
Long after the switch has been closed, what is the current in the 40 Ω resistor?
(a) A
(b) 0.3 A
(c) A
Long after switch is closed, potential across inductor = 0. Why???
Hence, current through 40 Ω resistor i = (3 V)/(10 Ω) = A (Par-V)

20. Fluxing Up The Inductor
How does the current in the circuit change with time? i
i(t)
Fast = Small τ
E/R
Slow= Large τ
Time constant of RL circuit: τ = L/R

21. RL Circuit Movie

22. Fluxing Down an Inductor
The switch is at a for a long time, until the inductor is charged. Then, the switch is closed to b.
What is the current in the circuit?
Loop rule around the new circuit walking counter clockwise: i
E/R
Exponential defluxing
i(t)

23. Inductors & Energy
Recall that capacitors store energy in an electric field
Inductors store energy in a magnetic field. i
P = iV = i2R
= power dissipated by R
Power delivered by battery
+ (d/dt) energy stored in L

24. Inductors & Energy Magnetic Potential Energy UB Stored in an Inductor.
Magnetic Power Returned from Defluxing Inductor to Circuit.

25. Example The switch has been in position “a” for a long time.
It is now moved to position “b” without breaking the circuit.
What is the total energy dissipated by the resistor until the circuit reaches equilibrium?
9 V
10 Ω
10 H
When switch has been in position “a” for long time, current through inductor = (9V)/(10Ω) = 0.9A.
Energy stored in inductor = (0.5)(10H)(0.9A)2 = 4.05 J
When inductor de-fluxes through the resistor, all this stored energy is dissipated as heat = 4.05 J.

26. E=120V, R1=10Ω, R2=20Ω, R3=30Ω, L=3H.
What are i1 and i2 immediately after closing the switch?
What are i1 and i2 a long time after closing the switch?
What are i1 and i2 immediately after reopening the switch?
What are i1 and i2 a long time after reopening the switch?

28. Energy Density in E and B Fields

29. The Energy Density of the Earth’s Magnetic Field Protects us from the Solar Wind!

30. Example, RL circuit, immediately after switching and after a long time: SP30.05

34. Example, RL circuit, during a transition: SP30.06

35. Example, Energy stored in a magnetic field: SP30.07