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r^ (7.9)

We now have a relation between the magnetic eld and the current that generates the eld,

hence the constant A in expression (5.13) is now determined. Note that the magnetic eld

depends only on the total current through the wire, but that is does not depend on

the distribution of the electric current density J within the wire as long as the electric

current density exhibits cylinder symmetry. Compare this with the result you obtained in

problem b of section (6.2)!

CHAPTER 7. THE THEOREM OF STOKES

64

I

B

Figure 7.5: Geometry of the magnetic eld induced by a current in a straight in nite wire.

7.4 Magnetic induction and Lenz's law

The theory of the previous section deals with the generation of a magnetic eld by a

current. A magnet placed in this eld will experience a force exerted by the magnetic

eld. This force is essentially the driving force in electric motors using an electrical

current that changes with time a time-dependent magnetic eld is generated that exerts

a force on magnets attached to a rotation axis.

In this section we will study the reverse e ect what is the electrical eld generated by

a magnetic eld that changes with time? In a dynamo, a moving part (e.g. your bicycle

wheel) drives a magnet. This creates a time-dependent electric eld. This process is called

magnetic induction and is described by the following Maxwell equation (see ref. 31]):

E = @B (7.10)

@t

r ;

To x our mind let us consider a wire with endpoints A en B, see gure (7.6). The

direction of the magnetic eld is indicated in this gure. In order to nd the electric eld

induced in the wire, integrate equation (7.10) over the surface enclosed by the wire

Z Z @B

(r E) dS = @t dS : (7.11)

;

S S

Problem a: Show that the right hand side of (7.11) is given by =@t, where is the

;@

magnetic ux through the wire. (See section (4.1) for the de nition of the ux.)

We have discovered that a change in the magnetic ux is the source of an electric eld.

The resulting eld can be characterized by the electromotive force FAB which is a measure

7.4. MAGNETIC INDUCTION AND LENZ'S LAW 65

^

n C

B

A

B

Figure 7.6: A wire-loop in a time-dependent magnetic eld.

of the work done by the electric eld on a unit charge when it moves from point A to point

B, see gure (7.6):

ZB

E dr :

FAB (7.12)

A

Problem b: Show that the electromotive force satis es

@:

FAB = (7.13)

@t

;

Problem c: Because of the electromotive force an electric current will ow through the

wire. Determine the direction of the electric current in the wire. Show that this

current generates a magnetic eld that opposes the change in the magnetic eld

that generates this current. You have learned in section (7.3) the direction of the

magnetic eld that is generated by an electric current in a wire.

What we have discovered in problem c is Lenz's law, which states that induction currents

lead to a secondary magnetic eld which opposes the change in the primary magnetic eld

that generates the electric current. This implies that coils in electrical systems exhibit

a certain inertia in the sense that they resist changes in the magnetic eld that passes

through the coil. The amount of inertia is described by a quantity called the inductance

L. This quantity plays a similar role as mass in classical mechanics because the mass of

a body also describes how strongly a body resists changing its velocity when an external

force is applied.

CHAPTER 7. THE THEOREM OF STOKES

66

7.5 The Aharonov-Bohm e ect

It was shown in section (4.3) that because of the absence of magnetic monopoles the

magnetic eld is source-free: (r B) =0. In electromagnetism one often expresses the

magnetic eld as the curl of a vector eld A:

B= A: (7.14)

r

The advantage of writing the magnetic eld in this way is that for any eld A the magnetic

eld satis es (r B) =0 because (r A) = 0.

r

Problem a: Give a proof of this last identity.

The vector eld A is called the vector potential. The reason for this name is that it plays

a similar role as the electric potential V . Both the electric and the magnetic eld follows

from V and A respectively by di erentiation: E = and B = A. The vector

; rV r

potential has the strange property that it can be nonzero (and variable) in parts of space

where the magnetic eld vanishes. As an example, consider a magnetic eld with cylinder

symmetry along the z-axis that is constant for r < R and which vanishes for r > R:

(

B = B0 ^

0z for r<R (7.15)

for r>R

see gure (7.7) for a sketch of the magnetic eld. Because of cylinder symmetry the vector

potential is a function of the distance r to the z-axis only and does not depend on z or '.

Problem b: Show that a vector potential of the form

A = f(r)^ (7.16)

'

gives a magnetic eld in the required direction. Give a derivation that f(r) satis es

the following di erential equation:

(

1 @ (rf(r)) = B0 for r < R (7.17)

for r > R

r @r 0

These di erential equations can immediately be integrated. After integration two integra-

tion constants are present. These constants follow from the requirement that the vector

potential is continuous at r = R and from the requirement that f(r = 0) = 0. (This

^

requirement is needed because the direction of the unit vector ' is unde ned on the z-axis

where r = 0. The vector potential therefore only has a unique value at the z-axis when

f(r = 0) = 0.)

Problem c: Integrate the di erential equation (7.17) and use that with the requirements

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