1. -Self Inductance -Inductance of a Solenoid -RL Circuit -Energy Stored in an Inductor
AP Physics C
Mrs. Coyle

2. Induced emf and induced current are caused by changing magnetic fields.

3. Self-Inductance
Switch Closed: current increases, creates an increasing magnetic field which in turn induces an induced emf
Switch Open: No current and no magnetic field

4. The direction of the induced emf is opposite the direction of the emf of the battery
Gradually the net current increases to an equilibrium value

5. This effect is called self-inductance
Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself
The emf εL is called a self-induced emf

6. Inductors
In general wires between resistors cause a small self inductance which is ignored.
However, some circuit elements such as solenoids can cause a significant inductance, L. These objects are called inductors.
Symbol for an inductor:

7. Inductance Induced emf:
L is a proportionality constant called the inductance of the coil and it depends on the geometry of the coil and other physical characteristics
The inductance is a measure of the opposition to a change in current

8. Inductance Units The SI unit of inductance is the henry (H)
Named for Joseph Henry

9. Inductance in a Solenoid (Coil)
The polarity of the induced emf, from Lenz’s Law, opposes the change in magnetic flux.

10. Inductance of a Solenoid
Assume a uniformly wound solenoid having N turns and length ℓ
Assume ℓ is much greater than the radius of the solenoid
The interior magnetic field is uniform:

11. Inductance of a Solenoid
(eq.1)
Faraday’s Law
Equate eq. 1 and 2 solve for L for a solenoid:
Remember also :
(eq.2)

12. Inductance of a Solenoid
The magnetic flux through each turn is
Note that L depends on the geometry of the object

13. RL Circuit (Resistor and Inductor)
Kirchhoff’s loop rule for:
Time Constant, t

14. RL Circuit Current The inductor affects the current exponentially
The current does not instantly increase to its final equilibrium value
If there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value as expected

15. RL Circuit, Time Constant ,t
t = L / R
When T=t,
I= Ieq,
then, the current has reached 63.2% of its equilibrium value.

16. LR Circuit, Charging
Prove:

17. LR Circuits, discharging
Prove:

18. Energy Stored in an Inductor, U
The rate at which energy is being supplied by the battery (Power=IV)
Remember for a capacitor:

19. Ex: RL Circuit What happens JUST AFTER the switch is closed?
What happens LONG AFTER switch has been closed?
What happens in between?
Note:
At t=0, a capacitor acts like a regular wire; an inductor acts like an open wire.
After a long time, a capacitor acts like an open wire, and an inductor acts like a regular wire.

20. Ex: RL Circuits
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 W
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 = E9 V

21. Ex: RL Circuits
Immediately after the switch is closed, what is the current i through the 10 W resistor?
(a) A
(b) 0.3 A
(c) 0
40 W
10 H
3 V
10 W
Immediately after switch is closed, current through inductor = 0.
Hence, current trhough battery and through 10 W resistor is i = (3 V)/(10W) = 0.3 A
Long after the switch has been closed, what is the current in the 40W resistor?
(a) A
(b) 0.3 A
(c) A
Long after switch is closed, potential across inductor = 0.
Hence, current through 40W resistor = (3 V)/(40W) = A

22. How does the current in the circuit change with time?
Ex: RL Circuits
How does the current in the circuit change with time? i
“Time constant” of RL circuit = L/R
i(t)
Small L/R
Large L/R t

23. Ex: Energy Stored in a B-Field
9 V
10 W
10 H
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?
When switch has been in position “a” for long time, current through inductor = (9V)/(10W) = 0.9A.
Energy stored in inductor = (0.5)(10H)(0.9A)2 = 4.05 J
When inductor “discharges” through the resistor, all this stored energy is dissipated as heat = 4.05 J.

24. Energy Density of a Magnetic Field
U = ½ L I 2
Aℓ is the volume of the solenoid
Magnetic energy density, uB :
This applies to any region in which a magnetic field exists (not just the solenoid)

25. Example 32.5: The Coaxial Cable
Calculate L for the cable
The total flux is
Therefore, L is
The total energy is

26. Prob.#3
A 2.00-H inductor carries a steady current of A. When the switch in the circuit is opened, the current is effectively zero after 10.0 ms. What is the average induced emf in the inductor during this time? Ans: 100V

27. Prob#5
A 10.0-mH inductor carries a current I = Imax sin ωt, with Imax = 5.00 A and ω/2π = 60.0 Hz. What is the back emf as a function of time? Ans: (18.8V)cos (377t)

28. Prob.# 7
An inductor in the form of a solenoid contains 420 turns, is 16.0 cm in length, and has a cross-sectional area of 3.00 cm2. What uniform rate of decrease of current through the inductor induces an emf of 175 μV? Ans: A/s

29. Prob.#14
Calculate the resistance in an RL circuit in which L = 2.50 H and the current increases to 90.0% of its final value in 3.00 s.

30. Prob.#16 Show that is a solution of the differential equation
where and Io is the current
at t=0.

31. Prob. #20
A 12.0-V battery is connected in series with a resistor and an inductor. The circuit has a time constant of 500 μs, and the maximum current is 200 mA. What is the value of the inductance?

32. Prob.# 24
A series RL circuit with L = 3.00 H and a series RC circuit with C = 3.00 μF have equal time constants. If the two circuits contain the same resistance R, (a) what is the value of R and (b) what is the time constant?

33. Prob.#26
12.0 W
12 V
1200 W
2.00 H
A) What is the current in the circuit a long time after the switch has been in position a?
B) Now the switch is thrown from a to b. Compare the initial voltage across each resistor and across the inductor.
C) How much time elapses before the voltage across the inductor drops to 12.0V?

34. Prob.#31
An air-core solenoid with 68 turns is 8.00 cm long and has a diameter of 1.20 cm. How much energy is stored in its magnetic field when it carries a current of A?

35. Prob.#33
On a clear day at a certain location, a 100-V/m vertical electric field exists near the Earth’s surface. At the same place, the Earth’s magnetic field has a magnitude of × 10–4 T. Compute the energy densities of the two fields.

36. Prob.# 36
A 10.0-V battery, a 5.00-Ω resistor, and a 10.0-H inductor are connected in series. After the current in the circuit has reached its maximum value, calculate (a) the power being supplied by the battery, (b) the power being delivered to the resistor, (c) the power being delivered to the inductor, and (d) the energy stored in the magnetic field of the inductor.