EE3321 ELECTROMAGENTIC FIELD

THEORYWeek 8

Faraday’s LawInductance

Energy

The law states that the induced electromotive force or EMF in any closed circuit is equal to the time rate of change of the magnetic flux through the circuit:

where

◦ Ɛ is the electromotive force (EMF) in Volts ◦ ΦB is the magnetic flux through the circuit in Webers

Faraday's Law of Induction

The magnetic flux ΦB through a surface S, is defined by an integral over a surface:

Magnetic Flux

Example

If the loop enters the field at a constant speed, the flux will increase linearly and EMF will be constant.

What happens when the entire loop is in the field?

Analysis

A conducting rod of length L moves in the x-direction at a speed vax as show below.

The wire cuts into a magnetic field B az. Calculate the induced EMF.

Exercise

Compute the total flux ΦB

ΦB = B A = B L (xo + vt)

Calculate the rate of change

Ɛ = - dΦB/dt = - vBL

Since the magnetic flux through the loop increases linearly the EMF induced in the loop must be constant.

Procedure

An EMF is induced on the rectangular loop of the generator.

The loop of area A is rotating at an angular rate ω (rads).

The magnetic field B is held constant.

AC Generator

Exercise

Assume B = Bo ax

the square loop rotates so its normal unit vector is an = ax cos ωt + ay sin ωt

Determine the flux as a function of time and the induced EMF

Inductance is the property in an electrical circuit where a change in the current flowing through that circuit induces an EMF that opposes the change in current.

Inductance

The term 'inductance' was coined by Oliver Heaviside. It is customary to use the symbol L for inductance, possibly in honor of Heinrich Lenz.

Other terms coined by Heaviside◦ Conductance G◦ Impedance Z◦ Permeability μ◦ Reluctance R

EE (Heaviside) Notation

The electric current produces a magnetic field and generates a total magnetic flux Φ acting on the circuit.

This magnetic flux tends to act to oppose changes in the flux by generating an EMF that counters or tends to reduce the rate of change in the current.

The ratio of the magnetic flux to the current is called the self-inductance which is usually simply referred to as the inductance of the circuit.

Inductance

Derive an expression for the inductance of a solenoid

Since B = μNI/ℓ Φ = BA = μNIA/ℓ

thus L = NΦ/I = μN2A/ℓ

Example

Derive an expression for the inductance of a toroid. Recall that

B = μNI 2πr

Exercise

The pickup from an electric guitar captures mechanical vibrations from the strings and converts them to an electrical signal.

The vibration from a string modulates the magnetic flux, inducing an alternating electric current.

Magnetic Pickups

Taking the time derivative of the flux linkage

λ = NΦ = Li

yields

= v

This means that, for a steady applied voltage v, the current changes in a linear.

Flux Linkage

Let the current through an inductor be

i(t) = Io cos(ωt).

Determine v(t) across the inductor.

Exercise

Ferromagnetic Core

In practice, small inductors for electronics use may be made with air cores.

For larger values of inductance and for transformers, iron is used as a core material.

The relative permeability of magnetic iron is around 200.

Ferromagnetic Materials

Mutual Inductance

Here, L11 and L22 are the self-inductances of circuit one and circuit two, respectively. It can be shown that the other two coefficients are equal: L12 = L21 = M, where M is called the mutual inductance of the pair of circuits.

Flux Coupling Consider for example two circuits carrying

the currents i1, i2. The flux linkages of circuits 1 and 2 are

given by

L12 = L21 = M

Describe what will happen when the switch is on.

Example

The energy density of a magnetic field is

u = B2

2μ

The energy stored in a magnetic field is

U = ½ LI2

Energy

In a certain region of space, the magnetic field has a value of 10-2 T, and the electric field has a value of 2x106 V/m.

What is the combined energy density of the electric and magnetic fields?

Exercise:

Read Sections 6-1, 6-2, 6-4, and 6-5. Solve end-of-chapter problems 6.2, 6.5,

6.10, and 6.12.

Homework