Lectures 13 and 14 Electromagnetic induction

Introduction

In the previous lectures we saw that when a current passes through a wire then a magnetic field is produced. It is reasonable to guess that there might be the reverse effect: if a wire is passed through a magnetic field then a current is produced. This effect, known as electromagnetic induction, was discovered in 1831-1832 during a series of experiments by Michael Faraday.

Faraday’s Experiments

· Mutual inductance: two coils A and B are arranged such that when a current flows in A some of the magnetic flux produced links (i.e. crosses) B. If the current through A changes then a current is induced in B.

· Relative motion: a coil is placed such that the magnetic flux from a source M (this may be a magnet or a current) links it. If relative motion occurs between the coil and the source such that the flux linking the coil changes then a current is induced in the coil.

· Cutting of flux by a conductor: when part of a circuit moves and, in doing so, cuts magnetic flux then a current is induced in the circuit.

· Electromotance (voltage) or current?: the induced current is proportional to the conductance of the circuit. Hence a given change produces a definite electromotance (or electromotive force e.m.f – the ability to drive current around a circuit – related to voltage) rather than a definite current.

· Self induction: an electromotance is induced in a circuit due to changes in its own current.

· The magnitude of the induced electromotance is proportional to the rate of change.

The results of Faraday’s experiments are described by his laws of electromagnetic induction, summarised as follows

An electromotance is induced when

1. A rigid stationary circuit is placed in a time varying magnetic field

2. A rigid circuit moves in a steady B-field such that the magnetic flux through the circuit changes

3. Part of a circuit moves and in doing so cuts magnetic flux

Effect 1 is sometimes known as transformer electromotance and effects 2 and 3 as motional electromotance.

Lenz’s Law

Faraday’s laws do not give the direction or sense of the induced electromotance or current. This is given by Lenz’s law which states ‘whenever a change produces an induced current the direction of flow of this current is such as to oppose the change causing it’.

Some definitions

Magnetic flux: This is defined in the same way as electric flux. If a magnetic field B passes through a surface A (described by a vector A) then the flux through A is F=B×A (i.e. it is the component of B normal to the surface multiplied by the area of the surface).

Electromotance (or electromotive force e.m.f) The electromotance (e) is defined as the energy or work done per unit charge (Q_t) when it is moved around a closed path L.

Electromotance =

(the final term results because F/Q_t=E)

The units of electromotance are the Volt (V).

Because for electrostatics , static charges are unable to produce an electromotance.

Motional electromotance

Consider the conductor in the figure which is moving with respect to the magnetic field B. Charges within the conductor experience a magnetic force given by F=Qv´B and hence an effective electric field E=v´B. If the conductor is part of a complete circuit then this E-field will cause a current to flow around the circuit.

Because this E-field results from the effect of a magnetic field and not static charges (and as we will see is capable of producing an electromotance) we denote it as E_M to distinguish it from electrostatic fields which we denote as E_S.

If the E-field E_M exists along an element dl of the moving conductor then an electromotance

de =E_M×dl= (v´B)×dl

results.

Because of the form of this result it is only the components of v and dl which are perpendicular to the B-field (v_┴ and dl_┴ respectively) which are important.

Hence the electromotance can be written in the non-vector form

de =v_┴Bdl_┴

For a complete circuit the total electromotance is given by the line integral

Motional electromotance in terms of flux cutting

The above result can also be expressed in terms of the rate of change of flux cutting. In time dt the element dl_┴ moves a distance v_┴dt and hence sweeps out an area v_┴dtdl_┴. The flux cut is hence this area multiplied by the field dF= v_┴dtdl_┴B.

Hence

but this is identical to the result for the motional electromotance derived previously so we have

(where F is the flux cut)

This result can be shown to be a general one which applies to all types of electromagnetic induction (not just motional). A negative sign is inserted in the equation to account for Lenz’s law

e (F is the flux linking the circuit)

This equation fully summarises Faraday’s law of electromagnetic induction.

Induced currents and charges

If the flux F through a circuit changes then an electromotanceE is produced and, from Ohm’s law, a current I=ℰ/R, where R is the resistance of the circuit, will flow.

If the flux through the circuit varies from an initial value F_i at time t_i to a final value F_f at time t_f (F_i-F_f=DF) then the total charge which flows in the circuit is

This result may be used in conjunction with a small coil ( a search coil) to determine the size of a magnetic field.

The differential form of Faraday’s law

Faraday’s law tells us that a new type of electric field (E_M) is produced in situations where the flux through a circuit changes.

Although so far we have considered the effects of E_M on conductors, E_M will be present even in the absence of these conductors (in a similar way to the existence of electrostatic fields (E_S) independent of the presence or not of any charges).

Hence the electromotance can be calculated around any arbitrary closed path L in free space

but the flux can be written as

where S is any surface enclosed by the path L. Hence

The order of integration and differentiation on the right hand side of this equation can be reversed and the left hand side can be transformed into a surface integral by applying Stoke’s theorem

Because the second and third terms contain an integral over the same surface S the arguments of the two integrals must be equal at any given point. Hence

At any point in space the total E-field E is the sum of E_S and E_M

E=E_S+E_M

Because Ñ´E_S=0 and Ñ´E=Ñ´E_S+Ñ´E_M we can write finally

This is the fourth Maxwell equation.

It can also be shown that the divergence of E_M is always zero.

Ñ×E_M=0

Hence Ñ×E=r/e₀ as for purely electrostatic fields.

Self-inductance

For any circuit the magnetic field at any point is proportional to the current I flowing in the circuit. Hence the magnetic flux F which links the circuit is also proportional to I. The constant of proportionality (which is a function of the shape and size of the circuit) is known as the self-inductance L.

F=LI

The unit of self-inductance is the henry (H).

A device designed to exhibit a specific value of self-inductance is known as an inductor.

When the current through a circuit changes an electromotance or voltage is produced, given by

the negative sign indicates that the induced voltage has a direction which opposes the change in I.

Calculation of self-inductance

A solenoid: It was shown in a previous lecture that for a helically wound solenoid with a length much greater than its diameter that the field within the solenoid was approximately constant and had a value m₀nI where n is the number of turns per unit length.

The flux across each turn of area A is hence

Am₀nI

and if the solenoid has a length l and hence nl turns, the total flux is

Am₀n²Il

Using the definition of L: L=F/I

Þ L= Am₀n²l

Magnetic energy

(a) In terms of self-inductance

The electromotance or voltage across an inductor is E=LdI/dt. Hence the rate at which energy is stored is in the inductor (Power = voltage x current)

=LIdI/dt

In time dt the energy stored by the inductor is LIdI so the total energy U_M stored when the current increases from 0 to I is

(b) In terms of the magnetic field

It was shown above that the self-inductance of a solenoid is L= Am₀n²l and the field inside the solenoid is B=m₀nI. Hence substituting in for L and using the formula for B to eliminate I the magnetic energy stored by the solenoid is