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
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 =
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
t
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.