Electromagnetism - Lecture 7: Inductance & AC Circuits, Exercises of Electromagnetism and Electromagnetic Fields Theory

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Electromagnetism - Lecture 7

Inductance & AC Circuits

Self Inductance

Mutual Inductance

Magnetic Energy

AC Circuits

Electrical Impedance

Resonance in LCR Circuits

1

Self Inductance

A current loop produces a dipole magnetic field with

m

=I

A

ˆn

Change in I causes change in

B

Change in Φ

B

through loop causes emf

E

that

opposes

change

E

(^) d Φ B

dt

L

(^) d I

dt

where the

self inductance

of the loop is: L =

d Φ B

d I

Self inductance is a purely

geometric

property of the loop

The unit of inductance is the Henry H = 1 Vs/A

2

Diagrams: Notes:

4

Inductances of Solenoids

For an infinite solenoid the magnetic field along the axis is:

B z = μ 0 n I

and the flux through

n

loops of radius

a

per unit length

l is:

Φ B = ∫ A B

.dS

μ 0 n I πa

2 nl

The self inductance of the solenoid is:

L =

d Φ B

d I = μ 0 n 2

πa

2 l

For two coaxial solenoids of radii

a

and

b < a

the mutual

inductance is:

M =

μ 0 n 1 n 2

πb

2 l

The flux coupling between the solenoids is

k

=

b/a

5

Magnetic Energy Density

The the magnetic field Magnetic energy is stored in an inductor through the creation of

energy density

of a magnetic field is proportional to the

square

of its amplitude:

dU

M

dτ

B

2

μ 0

vector potential and the current density: The energy density can also be expressed in terms of the magnetic

Φ B = ∮ L A

.dl

dU

M

dτ

J

2 A

A full derivation can be found in Grant & Phillips Pp.243-

7

Diagrams: Notes:

8

Electrical Impedance

Impedance is a generalization of the idea of resistance:

Z

I V

for an AC circuit containing any combination of L,

C

and

R

The change from cos

(^) ωt

to sin

(^) ωt

for

C

and L is represented by a

complex impedance

Z

R

= (^) R

Z

L

=

iωL

Z

C

=

−

i

ωC

amplitude and a phase between For a general circuit the impedance is a complex number with an

V

and I:

Z

Re

Z

(^) Im

Z

Z

R

(^) i ( Z C

(^) Z

L ) =

Z

0 e iφ

10

Combining Impedances

Sum of impedances in series:

Z = ∑ i Z i

R

R

1 (^) +

(^) R

2

L

L

1 (^) +

(^) L

2

C 1

C

1

C

2

Sum of impedances in parallel:

Z 1

i

Z

i

R 1

R

1

R

2

L 1

L

1

L 2 C = C 1 +

C

2

11

Impedance in LCR Circuits

The complex impedance of an LCR circuit is:

Z

R

(^) i ( ωL

ωC

The magnitude of this impedance is:

Z 0 = √ R 2 + (

ωL

ωC

2

and the current has the general form:

I

0 (^) cos

(^) φ

=

V

0 R

Z

(^02)

where

φ

is the phase angle between

V

and I:

tan

(^) φ

=

ωL

/ωC

R

13

Resonance in LCR Circuits

The minimum impedance is at the

resonance frequency

ω 0 :

Z

0

R

φ

= 0

ω 0 = (^) √

/LC

At the resonance

V

and I are in phase, and I is at a maximum

The power dissipated in an LCR circuit is averaged over one cycle:

< P >

I

V >

V

0 I 0 (^) cos

(^) φ

< P >

V

(^0 )

R

/R

2 ( ωL

/ωC

2 )

At resonance the maximum power is dissipated in the resistance:

< P >

V

(^0 )

R

14

Diagrams: Notes:

16