1. Physics Electricity and Magnetism Lecture 12 - Inductance, RL Circuits Y&F Chapter 30, Sect 1 - 4
Inductors and Inductance
Self-Inductance
RL Circuits – Current Growth
RL Circuits – Current Decay
Energy Stored in a Magnetic Field
Energy Density of a Magnetic Field
Mutual Inductance

2. Recap: Faraday’s Law of Induction
Magnetic Flux:
Faraday’s Law: A changing magnetic flux through a coil of wire induces an EMF in the wire, proportional to the number of turns, N.
Bind & iind oppose changes in FB
Lenz’s Law: The current driven by an induced EMF creates an induced magnetic field that opposes the flux change.
Induction and energy transfer: The forces on the loop oppose the motion of the loop, and the power required to sustain motion provides electrical power to the loop. i1 i2
Transformer principle: changing current i1 in primary induces EMF and current i2 in secondary coil.
Generalized Faraday Law: A changing magnetic flux creates non-conservative electric field.

3. conducting loop, resistance R
Changing magnetic flux directly induces electric fields
conducting loop, resistance R
Bind
A thin solenoid, n turns/unit length
Solenoid cross section A
Zero B field outside solenoid
Field inside solenoid:
Flux through the wire loop:
Changing current i  changing flux 
EMF is induced in wire loop:
Current induced in the loop is:
If di/dt is positive, B is growing, then Bind opposes change and i’ is counter-clockwise
B = 0 outside solenoid, so it’s not the Lorentz force
An electric field Eind is induced directly by dF/dt within the loop
Eind causes current to flow in the loop
Eind is there even without the wire loop (no current flowing)
Electric field lines are loops that don’t terminate on charge.
Eind is a non-conservative (non-electrostatic) field as the line
integral (potential) around a closed path is not zero
What makes the induced current I’ flow, outside solenoid?
Generalized Faradays’ Law
(hold loop path constant)

4. Example: Find the electric field induced by a region of changing magnetic flux
Find the magnitude E of the induced electric field at points within and outside the magnetic field.
Assume:
dB/dt = constant across the circular shaded area.
E must be tangential: Gauss’ law says any normal component of E would require enclosed charge.
|E| is constant on a circular integration path due to symmetry.
For r > R:
For r < R:
The magnitude of induced electric field grows linearly with r, then falls off as 1/r for r>R

5. Eind = - L di/dt L is the inductance
RELATED FARADAYS LAW EFFECTS IN COILS:
Mutual-induction between coils: di1/dt in “transformer primary” induces EMF and current i2 in “linked” secondary coil (transformer principle). N
Self-induction in a single Coil: di/dt produces “back EMF” due to Lenz & Faraday Laws. Find (or Bind) opposes dF/dt due to current change. Eind opposes di/dt.
ANY magnetic flux change is resisted (analogous to inertia)
Changing current in a single coil generates changing magnetic field/flux in itself.
Changing flux induces flux opposing the change, along with opposing EMF, current, and induced field.
The “back EMF” limits the rate of current (and therefore flux) change in the circuit
For increasing current, back EMF limits the rate of increase
For decreasing current, back EMF sustains the current
Inductance measures opposition to the rate of change of current
Eind = - L di/dt L is the inductance

6. Definition of Self-inductance
Joseph Henry
1797 – 1878
Recall capacitance: depends only on geometry
It measures charge stored per volt
Self-inductance depends only on coil geometry
It measures flux created per ampere
number of turns
flux through one turn depends on current & all N turns
self-inductance
cancels current dependence in flux above
SI unit of
inductance:
Why choose this definition?
Cross-multiply…….
Take time derivative
Another form of Faraday’s Law!
L contains all the geometry
EL is the “back EMF”

7. Example: Find the Self-Inductance of a solenoid+ - RL L
Field:
Flux in just
one turn:
Apply definition of self-inductance:
Depends on geometry
only, like capacitance.
Proportional to N2 !
N turns
Area A
Length l
Volume V = Al
where
ALL N turns contribute to self-flux through ONE turn
Note: Inductance per unit length has same dimensions as m0
Check: Same formula for L if you start with Faraday’s Law for FB:
for solenoid use above

8. Constant current implies induced voltage EL = 0
Example: calculate self-inductance L for an ideal solenoid l
Ideal inductor (ideal solenoid) (abstraction):
Non-ideal inductors have internal resistance: L
Vind r
resistance
with
wire a
like
behaves
Inductor ,
Constant current implies induced voltage EL = 0
current on
depends ir of
Direction
di/dt E
terminal voltage
measured V
IND = -

9. Lenz’s Law applied to Back EMF
If i is increasing:
EL opposes increase in i
Power is being stored in B field of inductor + - EL i
Di / Dt
If i is decreasing:
EL opposes decrease in i
Power is being tapped from B field of inductor EL - + i
Di / Dt
What if CURRENT i is constant?
EL = 0, inductance acts like a simple wire

10. Induced EMF in an Inductor
12 – 1: Which statement describes the current through the inductor below, if the induced EMF is as shown?
Rightward and constant.
Leftward and constant.
Rightward and increasing.
Leftward and decreasing.
Leftward and increasing.

11. means that current is increasing and to the right
Example: Current I increases uniformly from 0 to 1 A. in
0.1 seconds. Find the induced voltage (back
EMF) across a 50 mH (milli-Henry) inductance. i + - EL
means that current is increasing and to the right
Negative result means that induced EMF is opposed to both di/dt and i.
Apply:
Substitute:
Inductors, sometimes called “coils”, are common circuit components.
Insulated wire is wrapped around a core.
They are mainly used in AC filters and tuned (resonant) circuits.
Devices:

12. Inductors in Circuits—The RL Circuit
Basic series RL circuit:
A battery with EMF E drives a current around the loop
Changing current produces a back EMF or sustaining EMF EL in the inductor.
Derive circuit equations using Kirchhoff’s loop rule.
Convert to differential equations and solve
(as for RC circuits).
New for Kirchhoff rule:
When traversing an inductor in the same
direction as the assumed current insert:

13. NEW TERM FOR KIRCHHOFF LOOP RULE
Series LR circuit + - E i L R a b EL
Find time dependent behavior
Inductance & resistance + EMF
Use Loop Rule
NEW TERM FOR KIRCHHOFF LOOP RULE
Given E, R, L: Find i, EL, UL for inductor as functions of time
Growth phase, switch to “a”. Loop equation:
i through R is clockwise and growing: EL opposes E
As t  infinity, current is large & stable, di/dt  0
Back EMF EL 0, i  E / R,
L acts like an ordinary wire
At t = 0, rapidly growing current but i = 0, EL= E
L acts like a broken wire
Energy is stored in L & dissipated in R
Decay phase, switch to “b”, exclude E, Loop equation:
Energy stored in L will be dissipated in R
EL acts like a battery maintaining previous current
At t = 0 current i = E / R, unchanged, CW
Current begins to collapse
Current  0 as t  infinity – energy is depleted

14. LR circuit: decay phase solutionb EL + -
After growth phase equilibrium, switch from a to b, battery out
Current i0 = E / R initially still flows CW through R
Inductance tries to maintain current using stored energy
Polarity of EL reversed versus growth. Eventually EL 0
Loop Equation is :
Substitute :
Circuit Equation:
di/dt <0
during decay,
opposite to current
Current decays exponentially: t 2t 3t i0 i
First order differential equation with simple exponential decay solution
At t = 0+: large current implies large di / dt, so EL is large (now driving current)
As t  infinity: current stabilizes, di / dt and current i both  0
EMF EL and VR also decay exponentially:
Compare to RC circuit, decay

15. LR circuit: growth phase solution
Loop Equation is :
Substitute :
Circuit Equation:
Current starts from zero, grows as a saturating exponential.
First order differential equation again - saturating exponential solutions
As t  infinity, di / dt approaches zero, current stabilizes at iinf = E / R
At t = 0: back EMF opposes battery, current is small, di / dt is large,. t 2t 3t
iinf i
i = 0 at t = 0 in above equation  di/dt = E/L
fastest rate of change, largest back EMF
Back EMF EL decays exponentially
Voltage drop across resistor VR= -iR
Compare to RC circuit, charging

16. EL Example: For growth phase find back EMF EL as a function of time +
Use growth phase solution S - + i EL
At t = 0: current = 0
Back EMF is ~ to rate of change of current
Back EMF EL equals the battery potential
causing current i to be 0 at t = 0
iR drop across R = 0
L acts like a broken wire at t = 0 -E EL
After a very long (infinite) time:
Current stabilizes, back EMF=0
L acts like an ordinary wire at t = infinity

17. Current through the battery - 1
12 – 2: The three loops below have identical inductors, resistors, and batteries. Rank them in terms of current through the battery just after the switch is closed, greatest first.
I, II, III.
II, I, III.
III, I, II.
III, II, I.
II, III, I.
I II III.
Hint: what kind of wire (ordinary or broken) does L act like?

18. Current through the battery - 2
12 – 3: The three loops below have identical inductors, resistors, and batteries. Rank them in terms of current through the battery a long time after the switch is closed, greatest first.
I, II, III.
II, I, III.
III, I, II.
III, II, I.
II, III, I.
I II III.
Hint: what kind of wire (ordinary or broken) does L act like?

19. Summarizing RL circuits growth phase
Extra
Inductor acts like a wire.
Inductor acts like an open circuit.
When t / tL is large:
When t / tL is small: i = 0.
Capacitive time constant
Compare:
Inductive time constant
The current starts from zero and increases up to a maximum of with a time constant given by
The voltage across the resistor is
The voltage across the inductor is

20. Summarizing RL circuits decay phase
Extra
The switch is thrown from a to b
Kirchoff’s Loop Rule for growth was:
Now it is:
The current decays exponentially:
Voltage across resistor also decays:
VR (V)
Voltage across inductor:

21. Energy stored in inductors
Recall: Capacitors store energy in their electric fields
derived using p-p capacitor
Inductors also store energy, but in their magnetic fields
Magnetic Potential Energy – consider power into or from inductor
UB grows as current increases, absorbing energy
When current is stable, UB and uB are constant
UB decreases as current decreases. It powers
the persistent EMF during the inductor’s decay phase
The B field stores energy with or without a conductor
derived using solenoid

22. Sample problem: Energy storage in magnetic field
of an inductor during growth phase
a) At equilibrium (infinite time) how much energy is stored in the coil?
L = 53 mH
E = 12 V
b) How long (tf) does it take to store half of this energy?
take natural log of both sides

23. flux through one turn of coil 2 due to all N1
Mutual Inductance
Example: a pair of co-axial, flux-linked coils
di1/dt in the first coil induces current i2 in the
second coil, in addition to self-induced effects.
M21 depends on geometry only, as did L, C, and R
Changing current in primary (i1) creates varying
flux through coil 2  induced E field in coil 2
mutual inductance
number of turns
in coil 2
flux through one turn of coil 2 due to all N1
turns of coil 1
current in coil 1
Definition:
cross-multiply
time derivative
Another form of Faraday’s Law!
M21 contains all the
geometry
E2 is EMF induced
in 2 by 1
The linked flux is the same (smaller coil) if coils reverse roles, so…..
proof not obvious

24. determines the linkage results for mutual inductance:
Calculating the mutual inductance M
Outer coil 1 is a flat loop of N1 turns (not a solenoid)
PRIMARY
large coil 1
N1 turns
radius R1
SECONDARY
small coil 2
N2 turns
radius R2
Assume uniform B
Use arc formula with f=2p
Flux through one turn of flat inner coil 2
depends on B1 and smaller area A2
Current i1 in Loop 1 is changing:
smaller radius (R2)
determines the linkage
Summarizing
results for mutual inductance:

25. Air core Transformer Example: Concentric coils
OUTER COIL – IDEAL SOLENOID
Coil diameter D = 3.2 cm
nouter = 220 turns/cm = 22,000 turns/m
Current iouter falls smoothly from
1.5 A to 0 in 25 ms
Field uniform across outer coil is:
FIND INDUCED EMF IN INNER COIL DURING THIS PERIOD
Coil diameter d = 2.1 cm = .021 m, Ainner = p d2/4, short length
Ninner = 130 turns = total number of turns in inner coil
Direction: Induced B is parallel to Bouter which is decreasing
Would the transformer work if we reverse the role of the coils?