Lectures 17&18: InductanceLearning Objectives

• To understand and to be able to calculate Self-Inductance

• To be able to obtain an expression for the Energy Stored by an Inductor

• To understand and to be able to calculate Mutual-Inductance

Self InductanceWhen current in the circuit changes, the magnetic flux changes also, and a self-induced voltage appears in the circuit. This is a direct consequence of electromagnetic induction dealt with in the previous two lectures.

(a) Definition used to find L Suppose a current I in a coil of N turns causes a flux B to thread each turn

€

NφB ∝ IThe self-inductance L is defined by the equation

€

NφB = LI

Calculation of Self-Inductance

Example: the Self-Inductance of a Solenoid

€

B = μ0nI

€

NΦB = πR2Bnl = πR2μ0n2Il

€

L =NΦB

I= μ0n

2πR2l

(b) Definition that describes the behaviour of an inductor in a circuit

From Faraday’s Law of Induction

BNdt

d−=ε

dt

dIL−=ε

The SI unit for inductance V s A-1

This is commonly called the henry (H)

Hence in terms of the inductance and the current

Example: the Self-Inductance of a Toroid

a

b

A cross section of a toroid

Consider an elementary strip of area hdr

r

dr

h r

iNB

πμ20=

( )∫=∫=b

aB hdrBAdB. ∫=

b

a r

driNh

π

μ

20

a

biNhln

20

πμ

=

⎟⎠

⎞⎜⎝

⎛=

=a

bhN

i

NL B ln

2

20

π

μ

The Energy Stored by an Inductor

I increasing

ε

dt

dILVab =

dt

dILIIVP ab ==

The energy dU supplied to the inductor during an infinitesimal time interval dt is:

dILIdU ×=

a b

The total energy U supplied while the current increases from zero to a final value I is

22

1

0

LIIdILUI

∫ ==

Example: the energy stored in a solenoid

2

2

1LIU =

Energy per unit volume (magnetic energy density)

2202 2

1In

lR

UuB μ

π==

nIB 0μ=0

222

0 μμ B

In =

22202

1lIRn πμ=

0

2

2μB

uB =The equation is true for all magnetic field configurations

Mutual InductanceA changing current in coil 1 causes a changing flux in coil 2 inducing a voltage in coil 2:

dt

dN B2

22

−=ε

122 iN B ∝

12122 iMN B =

dt

diM 1

212 −=ε

Mutual InductanceIt can be proved that the same value is obtained for M if one considers the flux threading the first coil when a current flows through the second coil

2

11

1

22

i

N

i

NM BB

=

= (mutual inductance)

dt

diM

dt

diM 2

11

2 & −=−= εε(mutually induced voltages)

A Metal DetectorSinusoidally varying current

Parallel to the magnetic field of Ct

•If a current I is established through each of the N windings of an inductor, a magnetic flux B links those windings. The inductance L of the inductor is:

€

L =NφBI

•A changing current I in a coil induces a voltage, where:

dt

dIL−=ε

Review and Summary

• An inductor with inductance L carrying current I has potential energy U:

22

1LIU =

• This potential energy is associated with the magnetic field of the inductor. In a vacuum, the magnetic energy per unit volume is 0

2

2μB

uB =

•If two coils are near each other, a changing current in either coil can induce a voltage in the other. This mutual induction phenomenon is described by

dt

diM

dt

diM 2

11

2 & −=−= εε

where M (measured in henries) is the mutual inductance for the coil arrangement