INDUCTANCEPlan:

Inductance

Calculating the Inductance

Inductors with Magnetic materials

Inductor• Capacitor stores energy in an

electric field

• Inductor stores energy in magnetic field

Mutual Inductance

Flux in the loop 2 is proportional to current in the loop one.

12 I 1212 IM

Self Inductance

A changing current not only induces emf in the nearby

loops, it also induces an emf in the source loop itself.

• Current i produces B in the interior of the coil

• Changing current generates emf in the same coil.

• Units : volt-second/ampere (Henry)

dt

diL

dt

diLL

Calculating the Inductance

dt

Nd BL

)(

dt

Nd

dt

diL B )(

NB is called flux linkage

i

NL B

A geometrical property

Inductance of a solenoid

nIB 0n is number of turns per unit length.

BAnlN B )(

nIAnlN B 0)(

AnlL 2

0

nAnlI

N B0)(

AnlL 20

Exercise• Inductance of a closed packed coil

of 400 turns is 8.0 mH. Calculate the magnetic flux through the coil when the current is 5 mA.

i

NL B =1 x 10-7 Wb

• A circular coil has a 10.3 cm radius and consists of 34 closely wound turns of wire. An externally produced magnetic field of 2.62 mT is perpendicular to the coil.

• (a) If no current in the coil what is the number of flux linkages?

BNLinkages WbN B31097.2

Exercise

L = 7.88 x 10-4H

•(b)When the current in the coil is 3.77 A the net flux in the coil is found to vanish. Find the inductance of the coil.

i

NL B

Inductance of a Toroid

r

NiB

2

0

AdBB

drhr

iNB

2

0

drhr

iNB

2

0

a

biNhB ln

20

a

bhN

i

NL B ln

2

20

Is the current increasing or decreasing?

•Ans: Decreasing

What is the value of inductance if rate of change of current is 25

kA/s and the emf is 17 V.

• Ans: L = 6.8 x 10-4 H

dtdi

L

Magnetic materials in inductors

• Dielectric substance increases the capacitance

• Same way, filling the magnetic material can increase the inductance.

0BB m

m is the permeability of the material

0LL m

i

NL B

i

NBAL

i

ABNL m 0

Inductance increases by a factor of m

Inductors in series= 1 + 2

dtdI

dtdI

dtdI

21

21 LLLeq

Inductors in parallel• Current through each inductor will add to

give the total current.

21 iii

dt

di

dt

di

dt

di 21

21 LLLeq

21

111

LLLeq

A toroid having 5.20 cm square cross section and an inside radius of 15.3 cm

has 536 turns of wire and carries a current of 810 mA. Calculate the

magnetic flux through a cross section

a

bhN

i

NL B ln

2

20

h = 5.20 cm, a = 15.3 cm, N = 536

b = (5.2 + 15.3) cm,

• Ans : B = 1.32 x 10-6 Wb

N

iLB

Current through a 4.6 H inductor varies with time as shown. Calculate the induced emf in the time intervals

(a) T = 0 to t = 2 ms

• Ans: 1.6 x 104 V

dt

dIL

Ans: 2.3 x 104 V

(a) T = 5 to t = 6 ms

A coaxial cable has an inner wire of radius a and an outer metal sheath with inner radius b. A current I flows down

the inner wire and back through the sheath. What is the self inductance of a

unit length of the cable?

ldrr

Ib

a

2

0

a

bI

lln

20

Find the magnetic dipole moment of a spherical shell

Rd

RdRdq sin2

2rT

dqd

2sinsin2 RT

RdRd

The magnetic field in the interior of a solenoid has the value of 6.5 x 10-4 T when the solenoid is empty. When it is filled with

iron, the field becomes 1.4 T. Find the average magnetic moment of iron under these conditions. Number of atoms per unit volume

is given as 8.49 x 1023 atoms/m3

• Magnetic moment per atom is

0

0

BB

M

n

M

M = 1.11 x 106 A/m

= 1.4 B

Two long parallel wires, each radius a, whose centers are at a

distance d apart carry equal current in opposite directions. Neglecting the flux within the

wire themselves, find the inductance of a length l of such

a pair of wires..

x

x

.

d

a

d-a

ldyyd

I

y

Iad

a

2200

p

y

d-y

ldyyd

I

y

Iad

a

2200

aadad

a ydyIl lnln

20

a

adlL

ln0

An alternating current I0 cost flows down a straight wire. Which runs along the axis of a toroidal coil with rectangular cross section. The coil is connected to a resistance of R.

(a) What is the emf induced in the toroid?

Flux through the N turns of the toroid

hdrr

INb

a

2

0

hdrr

INb

a

2

0

dt

d

tIa

bNh

sinln2 00

Find the induced current in the resistor

• Calculate the back emf in the coil due to this induced current.

tIa

b

R

Nh

RIc

sinln2 0

0

dt

dIL c

b dt

dI

a

bhN cb

ln

2

20

In some circuits a spark occurs between the poles of the switch when the switch

is opened. Yet for the switch for the circuit is closed no spark occurs. Why is

this difference?

• When the switch is opened, the sudden drop in B induced an emf that attempts the original current flowing. This causes a spark as the current bridges the air gap between the poles of the switch.