The current through the inductor can be considered a sum of the current in the circuit and the induced current. The current in the circuit will be constant, but the induced current will decrease exponentially to zero.

RtLI e

R R

Current in circuit, once switch is closed

Induced current

The induced current and the circuit current are in opposite directions

0 inducedI I I

The exponential growth of the current through the inductor occurs at a defined rate according to the time constant of the circuit.

L

R This is the time constant for

a RL circuit.Do not confuse the time constant for an RC circuit with that for an RL circuit!

1RtLI e

R

0 1

t

I I e

Same form as the charge in an RC circuit.

LR

S

We can also examine the situation for a decreasing current.

0dI

L IRdt

Define:

x Idx dI

0 0x IR

0L dI

IR dt

0L dx

xR dt

dx R

dtx L

0 0

x t

x

dx Rdt

x L

0

lnx R

tx L

0

RtLx x e

0

RtLI I e

0

t

I I e

How the current through the inductor changes for a decreasing circuit current.

When the switch is closed, the current through the circuit exponentially approaches a value I = / R. If we repeat this experiment with an inductor having twice the number ofturns per unit length, the time it takes for the current to reach a value of I / 2

1. increases.2. decreases.3. is the same.

1RtLI e

R

L

R

• When an inductor is connected to a circuit the battery must supply extra energy in order to overcome the back emf when a circuit is turned on.

• This extra energy is stored in the magnetic field of the inductor.• The amount of energy stored in the inductor can be determined by looking at the rate

that energy is stored by the inductor.

LP I dU dII L

dt dt

dU ILdI0 0

U I

dU ILdI 21

2U LI Energy stored in the magnetic field of the inductor.

It is sometimes more convenient to look at the energy density.

20N A

L

2

0N AL

20L n A

20L n V

212

B

LIUu

V V

2 20

2

n V I

V

2 2

0

1

2n I

0B nI

0

BI

n

For a solenoid

22

0 2 20

1

2B

Bu n

n

2

02B

Bu

Magnetic energy

density

This is valid for any shape inductor, not just a solenoid.

Example: In the following circuit switch S is closed at t = 0.a) What is the time constant for this circuit?b) What is the current in the circuit after 0.1 s?c) What is the self-induced voltage from the inductor after 0.1 s?d) How much energy is stored in the inductor after 0.1 s?

LR

S

e = 10 VR = 100 WL = 5 H

a) L

R 0.05s

b) 1t

I eR

86.5mA

c)L

dILdt

1td

L edt R

1 t

L

Le

R

tLe

R

1.35V

d)21

2U LI 18.7mJ

When we have two solenoids placed with their long axis coinciding, we know that if there is a changing flux the first solenoid will induce a voltage in the second and that the first solenoid will induce a voltage in itself.

The induced voltage in the second coil will cause a current in the second coil. What will we get from this current? A magnetic field!

Where will this magnetic field go? Back through the first solenoid!

The induced current in the second solenoid will create an induced magnetic field which will pass through the first solenoid inducing another current in the first solenoid.

We call this Mutual Induction. The two solenoids induce currents in each other.

When you are discussing the current in a circuit with multiple inductors you must include contributions from:

• The supply current• The self induced current in each inductor• The mutual induction between the two inductors

To determine the mutual inductance between two inductors let us consider two solenoids.

xxxxx

I

A

11 0 1

1

NB I

22 0 2

2

NB I

22 2

dN

dt

2 1 2

dN B Adt

12 2 0 1

1

NdN A I

dt

0 1 2 2 12

1

N N A dI

dt

12 12

dIM

dt

This is the mutual inductance (M) between the two solenoids. Only depends on geometry!

11 1

dN

dt

1 2 1

dN B Adt

21 1 0 2

2

NdN A I

dt

0 2 1 1 21

2

N N A dI

dt

21 21

dIM

dt

M12 and M21 will always be the same for the mutual inductance between two inductors. It is more common to replace M12 and M21 with M.

Same

Electricity and MagnetismMagnetic Circuits

CH 30: Alternating Current Circuits

A second possible circuit configuration is using an inductor combined with a capacitor or an LC circuit.

We are going to look at the case where we have only an inductor in series with a capacitor. The capacitor is fully charged.

C L

SWhat happens when the switch is closed?

The capacitor discharges, releasing energy into the circuit. This creates a sudden change in the magnitude of the current flowing through the circuit from the capacitor. The current from the capacitor will then decrease to zero.

What does the inductor do?

The sudden increase in current from the capacitor corresponds to a large induced current (which gives a net current of zero). Energy is stored in the magnetic field of the inductor. The slower decrease in the current from the capacitor will cause an induced current to be generated in the inductor in the same direction as the current flowing out of the capacitor. This corresponds to the release of energy from the collapse of the magnetic field of the inductor.

I

The current now flows into the opposite plates of the capacitor, which is stored in the electric field of a now oppositely polarized capacitor. The capacitor will then begin to discharge.

For an ideal LC circuit this process would continue indefinitely.

IinducedI