4Electrodynamics

IB Electromagnetism

4.3 Resistance

The story so far is that we change the flux, an emf is produced, and charges

are accelerated. In principle, we should be able to compute the current. But

accelerating charges are complicated (they emit light). Instead, we invoke a new

effect, friction.

In a wire, this is called resistance. In most materials, the effect of resistance

is that

E

is proportional to the speed of the charged particles, rather than the

acceleration.

We can think of the particles as accelerating for a very short period of time,

and then reaching a terminal velocity. So

Law (Ohm’s law).

E = IR,

Definition (Resistance). The resistance is the R in Ohm’s law.

Note that

E

=

R

E ·

d

r

and

E

=

−∇φ

. So

E

=

V

, the potential difference.

So Ohm’s law can also be written as V = IR.

Definition

(Resistivity and conductivity)

.

For the wire of length

L

and cross-

sectional area A, we define the resistivity to be

ρ =

AR

L

,

and the conductivity is

σ =

1

ρ

.

These are properties only of the substance and not the actual shape of the

wire. Then Ohm’s law reads

Law (Ohm’s law).

J = σE.

We can formally derive Ohm’s law by considering the field and interactions

between the electron and the atoms, but we’re not going to do that.

Example.

d

z

x

y

Suppose the bar moves to the left with speed

v

. Suppose that the sliding bar

has resistance

R

, and the remaining parts of the circuit are superconductors

with no resistance.

There are two dynamical variables, the position of the bar

x

(

t

), and the

current I(t).

If a current I flows, the force on a small segment of the bar is

F = IB

ˆ

y ×

ˆ

z

So the total force on a bar is

F = IB`

ˆ

x.

So

m¨x = IB`.

We can compute the emf as

E = −

dΦ

dt

= −B` ˙x.

So Ohm’s law gives

IR = −B` ˙x.

Hence

m¨x = −

B

2

`

2

R

˙x.

Integrating once gives

˙x(t) = −ve

−B

2

`

2

t/mR

.

With resistance, we need to do work to keep a constant current. In time

δt

,

the work needed is

δW = EIδt = I

2

Rδt

using Ohm’s law. So

Definition

(Joule heating)

.

Joule heating is the energy lost in a circuit due to

friction. It is given by

dW

dt

= I

2

R.