1. Inductors. Stored energy. LR circuit.
Lecture 27
Inductor
Inductors. Stored energy LR circuit.
Spark plug

2. Induction between two coils
Change the current in coil 1
changes the B-flux through coil 2
induces an emf in coil 2

3. Mutual inductance M21 mutual inductance indicates how large a magnetic
flux through coil 2 due to current
in coil 1
Of course it works both ways:

4. Mutual inductance depends on the geometry and orientation of the coils
Mutual inductance depends on the geometry and orientation of the coils. It can be shown that
Therefore, we have:
Units: SI Henry 1 H = 1 Wb/A

5. ACT: Mutual inductance
If the current in coil 1 is as shown, which of the graphs gives the correct emf in coil 2? t i1 ε2 t t ε2 t ε2 A B C

6. Example: M for two solenoids
N2 turns, length l
N1 turns, length l
Craig Ogilvie

7. Self-inductance di/dt Ideal coil (no resistance)
If current through coil changes,
➝ flux through coil changes
➝ in each loop there is an induced emf
➝ loops “in series”
➝ emf induced between two ends of coil
The effect is called self-induction
Single coil is called an inductor

8. ACT: Inductor
In the circuit shown, the voltage in the power supply is turned up, so that the current increases. During this operation, which point, y or z, is at a higher potential? y z
Both have the same potential
By Lenz law, induced emf tries to oppose change. Current is increasing so emf tries to reduce it. Imagine a “battery” with
positive terminal near y.
ein

9. ACT: Inductor II
In the circuit shown, the voltage in the power supply has been on for a long time. The switch is then opened, as shown. Right after this, which point, a or b, is at a higher potential? a b
Both have the same potential
By Lenz law, induced emf tries to oppose change. Current is decreasing so emf tries to increase it. Imagine a “battery” with
positive terminal near a.
ein

10. In-class example: Inductance of a solenoid
A solenoid is constructed out of 1000 loops of wire wound around a 1.0 cm2 cross section tube that is 5 cm long. A 20-A current is run through the solenoid. What is the solenoid’s self inductance L?
length l, N turns
250 H
25 H
2.5 H
2.5 × 10−3 H
2.5 × 10−5 H
Inductance of a solenoid

11. Inductors against di/dt
Inductor always acts to oppose current change.
➝ generating a current in a circuit with an inductor will take some time.
➝ used in circuit design to protect against rapid changes of currents or spikes

12. RL circuits: current growth
Switch moved to position “a” at t = 0
drop in potential from
top of inductor to bottom
induced
emf
Kirchhoff’s law
Differential equation for i
+ initial condition
Qualitatively:

13. ε/R i t
If L is large, τ is large, i.e. current grows slowly.

14. In-class example: RL circuit
You have 1000 Ω resistor and you want to build an LR series circuit that will increase the current from 0 to 0.9 V0/R in 1 ms once the switch is closed. What value should L have? L R V0
1 H
0.43 H
0.1 H
0.043 H
0.01 H

15. ACT: Three circuits A. i1 > i2 > i3 initially
All batteries, inductors, resistors are identical.
Rank the circuits according to the current through battery just after switch is closed 1 2 3
A. i1 > i2 > i3 initially
B. i2 > i3 > i1 initially
C. i3 > i2 > i1 initially
Initially i = 0 through inductors.
➝ i1 = 0
R2 < R3 ➝ i2 > i3

16. ACT: Current decay A B C D
DEMO:
RL circuit
The switch has been in position “a” for a long time, then it is moved to position “b” at t = 0. What is the graph for current in the resistor?
induced
emf i t A B C D

17. Inductors store energy
After the switch is opened, the inductor drives the current through the resistor.
The energy dissipated in the resistor must have been stored in the inductor.

18. Energy stored in an inductor
To establish current (from i = 0 to i = I ):
Voltage across inductor:
(ignoring signs)
1) rate of energy supplied to inductor
2) increase of energy within inductor
3) energy stored within inductor

19. Using the stored energy
DEMO:
Spark
Energy is stored in L after switch is moved to “a” position
Switching to “b” releases energy stored in inductor
Time it takes for i = I to i = 0 can be very short
Energy released can cause an arc across switch contacts
pulling plug from wall-socket, spark (wires as the inductor)
spark plug, current was stored in ignition coil (an inductor)

20. Spark plug