self inductance

Self InductanceSelf Inductance

A variable power supply is connected to a loop. The current in the loop creates a magnetic field.

What happens when the power supply dial is turned down reducing the current, or turned up increasing the current?

Self InductanceSelf Inductance

The current does not go from zero to ε/R in the circuit immediately after the switch is closed:

1. as the current flows through, magnetic flux through the loop is set up 2. this is opposed by induced emf in the loop which opposes the change in net magnetic flux 3. by Lentz’s law, the induced E-field opposes the current flow

Self InductionSelf Induction

R

As a time-varying current flow through the conductor, the same thing happens:

1. as the changing current flows through, the magnetic flux through the loop changes 2. this is opposed by induced emf in the loop which opposes the change in net magnetic flux 3. by Lentz’s law, the induced E-field opposes the current flow

Self InductionSelf Induction

R

(From the Biot-Savart Law)

The self-inductance L is the proportionality constant. It depends on the shape of the loop, that is, its geometry.The self-induced ems is proportional to the rate of change of the current:

Unit for L: 1 henry (H) = 1 (V•s)/A

B

B

L

B IB Id dIdt dt

L

dIε L

dt (definition of L)

““Self-Inductance in a closed Self-Inductance in a closed loop”loop”

We must keep the shape and size of the loop fixed.

LI N

L

dI dL N

dt dt

An equivalent definition of L uses the integral of the above where is the flux through each loop produced by current I in each loop:

From Faraday’s Law for N loops and our definition of inductance, L:

To find L we use: L N

I

Quiz

A flat circular coil with 10 turns (each loop identical) has inductance L1 . A second coil, of the same size, shape and current passing through the conductor but with 20 turns, would have inductance:

A) 2 L1 B) ½ L1 C) 4 L1 D) ¼ L1 E) L1

Also:

We can see that inductance is the measure of the opposition to the change in current, I.

(Recall that resistance, R, is the opposition to current, I: R=V/I)

dtdIL

dt

dIL

/

Example 1: Long solenoidExample 1: Long solenoid

B

Given:N = 300 turnsr = 1 cml = 25cmCalculate L

I I l

2r

Solution

Example 2: Long solenoidExample 2: Long solenoid

B

Given:N = 300 turnsr = 1 cml = 25cmdI/dt = -50 A/s

Calculate the self-induced emf

I I l

2r

Solution

ab

Ir

Example 3: Self-Inductance of a coaxial Example 3: Self-Inductance of a coaxial cablecable

In the gap (a < r < b):

abL

r

IB

ln

o

o

2

2

Show that

Solution

Kirchhoff’s Loop Rule (again)Kirchhoff’s Loop Rule (again)

- resistor,

- capacitor,

- inductor,dtdI

LV

CqV

IRV

I

q q

I

Voltage change in going along path from left to right:

Path direction same as current

12 V20mH

What is the current 2 ms after the switch is closed?

Hint: write down Kirchoffs loop rule and solve the differential equation for I

Example 4

EnergyEnergy

εL= LdI/dt

+

-

How much work is done by an external emf to increase the current from 0 to Ifinal ?

I

0dI

Ldt

Kirchoff’s law gives:(Recall, P=VI)

dII LI

dt

Power supplied by external emf =

power absorbed by inductor

Power absorbed by inductor is:

221 LIU L

212 f

0

I f

LU L I dI LI

LdU LIdI

The total potential energy stored is:

LdU dI

LIdt dt

Find the potential energy of an inductor with L=400mH anda final current of 10A.

Examples of Inductors