1. Eddy Current
A current induced in a solid conducting object, due to motion of the object in an external magnetic field.
The presence of eddy current in the object
results in dissipation of electric energy
that is derived from mechanical motion
of the object.
The dissipation of electric energy in turn
causes the loss of mechanical energy of
the object, i.e., the presence of the field
damps motion of the object.

2. Self-Inductance, L Self-induction (henry) Faraday’s Law:
As current i through coil increases, magnetic flux through coil increases.
This in turn induces counter EMF in the coil itself (Lenz)
When current i is decreasing, EMF is induced again in the coil itself in such a way as to slow the decrease maintain the field.
Self-induction
(if flux linked)
(henry)
Faraday’s Law:

3. DEMO SELF INDUCTANCE 6D-10

4. Inductance and Faraday’s Law
(henry) + -
INERTIA
compare with
This could have been a battery, instead. With the simplifying advantage that the applied ε is then ~constant.

5. Qualitative discussion of currents in L
Note that if the initial current in an inductor, L, is ZERO, then it cannot instantly jump to a finite value when a potential difference is applied to it. Current I will “accelerate” from zero in a continuous way, just as the velocity of a mass at rest will accelerate in a continuous way under the influence of a force.
At large times, the transients have died out, and since I is no longer changing, no voltage drop exists across the inductor L 5

6. Reading Quiz 1 Which of the following statements is incorrect?
A| The inductance of a coil with N turns is proportional to N2 turns.
B| Like a capacitance (C) an inductor (L) depends on geometric factors and the nature of the material inside C and L.
C| The potential across an inductor depends only on the magnitude
of the current through the inductor.
D| The magnetic energy stored in an inductor is proportional to the
square of the current through the inductor.

7. Solenoid: Archetypical Inductor
Current i flows through a long solenoid of radius r with N turns in length l
For each turn
For the solenoid or
Al is the volume of the solenoid
Inductance, like capacitance, only depends on geometry (if made of conductor and air). Add a magnetically polarizable material, increase L, just as adding an electrically polarizable material (dielectric) increases C.

8. Potential Difference Across InductorΔV +V
V=0 I o
internal resistance must be symbolically separated from L !! L
“Analogous” to a battery
An ideal inductor has r =0
All dissipative effects are to be included in the external + internal resistances (i.e., including those of the iron core if any)

9. Energy Stored By Inductor
Switch on at t=0
As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. + -
Loop Rule:
3. Multiply through by I
Rate at which energy is stored in inductor L
Rate at which battery is supplying energy
Rate at which energy is dissipated by the resistor

10. 6C07 ENERGY STORED IN AN INDUCTOR

11. Where is the Energy Stored?
Energy must be stored in the magnetic field!
Energy stored by a capacitor is stored in its electric field
Consider a long solenoid where
area A
length l
So energy density of the magnetic field is
(Energy density of the electric field)

12. RL Circuits – Starting Current
Switch to ε at t=0
As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. + -
Loop Rule:
3. Solve this differential equation
τ=L/R is the inductive
time constant

13. No fully correct answer is offered See next slide
Warm-up quiz 2
The circuit is turned on at t= Which of the following statements is correct? R1 R2 L V
A| At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V/R1
B| At t = 0, the potential drop across the inductor is 0; When t = ∞, the current through R1 is V.
C| At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V/(R1+R2)
D| At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V/R2 X
yes
No fully correct answer is offered
See next slide 13

14. Warm-up quiz 2
The circuit is turned on at t= Which of the following statements is correct? R1 R2 L V
Analysis: At t=0, no current has yet started flowing in L.
BUT: Resistor R2 “short-circuits” L, in the sense that resistors are NON-INDUCTIVE and current can “instantly” start flowing in them.
So there IS an instant current path available at t=0, and we have a simple resistive voltage divider chain. VL = V R2/(R1+R2)
At t= I is no longer changing, so L acts like a wire and completely short- circuits R2. All the voltage drop is across R1, and the current is V/R1, as in the SECOND part of answer (a) 14

15. GROWTH AND DECAY OF CURRENT OF AN RL CIRCUIT 6C-05

16. Remove Battery after Steady I already exists in RL Circuits
Initially steady current Io is flowing: - +
Switch from e to f at t=0, causing back EMF to oppose the change.
Loop Rule:
Solve this differential equation
I cannot instantly become zero!
Self-induction
like discharging a capacitor

17. Starting Current through Inductor vs Charging Capacitor
Note: when “charging”, in L voltage leads current, in C, current leads voltage

18. Qualitative discussion of currents in L
Note that if the initial current in an inductor, L, is ZERO, then it cannot instantly jump to a finite value when a potential difference is applied to it. Current I will “accelerate” from zero in a continuous way, just as the velocity of a mass at rest will accelerate in a continuous way under the influence of a force.
At large times, the transients have died out, and since I is no longer changing, no voltage drop exists across the inductor L

19. Behavior of Inductors Increasing Current Decreasing Current
Initially, the inductor behaves like a battery connected in reverse.
After a long time, the inductor behaves like a conducting wire.
Decreasing Current
Initially, the inductor behaves like a “reinforcement” battery.

20. The switch in this circuit is initially open for a
Physics 241 March 22, :30 –Quiz 3
The switch in this circuit is initially open for a
long time, and then closed at t = 0. What is the
magnitude of the voltage across the inductor
just after the switch is closed?
zero V
R / L
V / R 2V

21. Physics 241 March :30 Quiz 3
The switch in this circuit is closed at t = 0. What is the magnitude of the voltage across the resistor a long time after the switch is closed?
zero V
R / L
V / R 2V

22. Physics 241 March 22, :30–Quiz 3
The switch in this circuit has been open for a long time. Then the switch is closed at t = 0. What is the magnitude of the current through the resistor immediately after the switch is closed?
zero
V / L
R / L
V / R
2V / R