4Electrodynamics

IB Electromagnetism

4.1 Induction

We’ll explore the Maxwell equation

∇ × E +

∂B

∂t

= 0.

In short, if the magnetic field changes in time, i.e.

∂B

∂t

6

= 0, this creates an

E

that accelerates charges, which creates a current in a wire. This process is called

induction. Consider a wire, which is a closed curve C, with a surface S.

We integrate over the surface S to obtain

Z

S

(∇ × E) · dS = −

Z

S

∂B

∂t

· dS.

By Stokes’ theorem and commutativity of integration and differentiation (assum-

ing S and C do not change in time), we have

Z

C

E · dr = −

d

dt

Z

S

B · dS.

Definition (Electromotive force (emf)). The electromotive force (emf) is

E =

Z

C

E · dr.

Despite the name, this is not a force! We can think of it as the work done on a

unit charge moving around the curve, or the “voltage” of the system.

For convenience we define the quantity

Definition (Magnetic flux). The magnetic flux is

Φ =

Z

S

B · dS.

Then we have

Law (Faraday’s law of induction).

E = −

dΦ

dt

.

This says that when we change the magnetic flux through

S

, then a current

is induced. In practice, there are many ways we can change the magnetic flux,

such as by moving bar magnets or using an electromagnet and turning it on and

off.

The minus sign has a significance. When we change a magnetic field, an emf

is created. This induces a current around the wire. However, we also know that

currents produce magnetic fields. The minus sign indicates that the induced

magnetic field opposes the initial change in magnetic field. If it didn’t and the

induced magnetic field reinforces the change, we will get runaway behaviour and

the world will explode. This is known as Lenz’s law.

Example. Consider a circular wire with a magnetic field perpendicular to it.

If we decrease

B

such that

˙

Φ <

0, then

E >

0. So the current flows anticlockwise

(viewed from above). The current generates its own

B

. This acts to increase

B

inside, which counteracts the initial decrease.

This means you don’t get runaway behaviour.

There is a related way to induce a current: keep B fixed and move wire.

Example.

d

Slide the bar to the left with speed

v

. Each charge

q

will experience a Lorentz

force

F = qvB,

in the counterclockwise direction.

The emf, defined as the work done per unit charge, is

E = vBd,

because work is only done for particles on the bar.

Meanwhile, the change of flux is

dΦ

dt

= −vBd,

since the area decreases at a rate of −vd.

We again have

E = −

dΦ

dt

.

Note that we obtain the same formula but different physics — we used Lorentz

force law, not Maxwell’s equation.

Now we consider the general case: a moving loop

C

(

t

) bounding a surface

S(t). As the curve moves, the curve sweeps out a cylinder S

c

.

S(t + δt)

S(t)

S

c

The change in flux

Φ(t + δt) − Φ(t) =

Z

S(t+δt)

B(t + δt) · dS −

Z

S(t)

B(t) · dS

=

Z

S(t+δt)

B(t) +

∂B

∂t

· dS −

Z

S(t)

B(t) · dS + O(δt

2

)

= δt

Z

S(t)

∂B

∂t

· dS +

"

Z

S(t+δt)

−

Z

S(t)

#

B(t) · dS + O(δt

2

)

We know that

S

(

t

+

δt

),

S

(

t

) and

S

c

together form a closed surface. Since

∇ · B = 0, the integral of B over a closed surface is 0. So we obtain

"

Z

S(t+δt)

−

Z

S(t)

#

B(t) · dS +

Z

S

c

B(t) · dS = 0.

Hence we have

Φ(t + δt) − Φ(t) = δt

Z

S(t)

∂B

∂t

· dS −

Z

S

c

B(t) · dS = 0.

We can simplify the integral over S

c

by writing the surface element as

dS = (dr × v) δt.

Then B · dS = δt(v × B) · dr. So

dΦ

dt

= lim

δ→0

δΦ

δt

=

Z

S(t)

∂B

∂t

· dS −

Z

C(t)

(v × B) · dr.

From Maxwell’s equation, we know that

∂B

∂t

= −∇ × E. So we have

dΦ

dt

= −

Z

C

(E + v × B) dr.

Defining the emf as

E =

Z

C

(E + v × B) dr,

we obtain the equation

E = −

∂Φ

∂t

for the most general case where the curve itself can change.