There's more than one way to compute the field of a moving charge.
On this page we're going to do it by transforming the charge's
field from its own rest frame, which we already know, into the frame of
an observer moving relative to the charge.
 Figure 1  Moving charge and observer:

We start with a charge,
q, and orient the axes such that
the charge is at the origin, at time 0, and is traveling parallel to the
x axis at velocity
v in the coordinate system
t, x, y, z for the inertial frame
S. In the charge's rest frame, frame
T, we'll use the coordinates τ,
x',
y',
z'. As usual we'll set
c=1, and we'll be using
cgs units, and metric signature (1,1,1,1). We'll also define
(1)
We want to find the field seen by an observer at time 0, stationary in frame
S, at location (
x,y,0).
In the charge's rest frame, frame
T, the field is static, so we don't need to worry about what time it is. At time 0 in frame
S, the point we're concerned with has the following coordinates in frame
T:
(
2)
The
E field at that point, in frame
T, will be:
(
3)
In the charge's rest frame there is, of course, no magnetic field. The Faraday tensor for frame
T will then be:
(4)
The Lorentz transform from
S to
T will be
(5)
and the Faraday tensor in frame
S will be
(6a)
which, just to be painfully clear, we write out "longhand" as
(6b)
Multiplying out, that's
(7)
from which we can read off the
E and
B field components in frame
S. Plugging in the values from (
3), we have:
(8) 
This deserves a couple of comments. First of all, we see that
(9)
which means the electric field points directly toward the moving charge  it does
not
"lag" the charge's position due to its motion. (Of course, if the
charge is accelerating it's a different story  the information that
the charge has changed its motion can't get to the observer until time
r/c, and until that moment the field will point to where the charge
would have been had it continued in a straight line at constant velocity.)
Second, the field isn't spherically symmetric. At points on the
x axis (
y=0), we have
(10a)
while at points along the
y axis (
x=0) we have
(10b)
The
field is "pancaked"  it's stronger out to the sides, and weaker in
front and in back of the charge. If we view the field as a
"bubble" around the charge, we can view the "bubble" as being Lorentz
contracted. However, Gauss's law still holds and the integral of
the field strength over the surface of a sphere around the charge is
the same in all inertial frames. So, if the "bubble" is
contracted foretoaft, it must bulge out to the sides, which isn't
quite the same as physical Lorentz contraction.
The BiotSavart Law of Magnetostatics
We'll now present a brief (and highly
unsatisfactory) derivation of the BiotSavart law.
Within a wire, we have
(B.1)
Then we can rewrite B
_{z} from (
8) as
(
B.2)
For slowly moving charges, where v << c, we can rewrite (
B.2) as
(
B.3)
where θ is as shown in
figure 1. If we define
(B.4)
then we can rewrite (B.3) as
(B.5)
Recalling that B
_{x}=B
_{y}=0, and with the observation that B
_{z}
is perpendicular to the current and radius vectors, and with an
application of the righthand rule to check that the direction of the
vector is correct, we can therefore write
(
B.6)
We can now integrate it over all currents, to obtain the
total field due to the current in an arbitrary collection of wires; this is the BiotSavart law:
(B.7) 
Unfortunately our derivation is unsatisfactory for a couple of reasons. First, I eliminated
γ in (
B.3) by assuming currents were moving much slower than
c.
This is normally the case, but just the same we'd like to have
some idea of whether this law applies to, say, the beam in a particle
accelerator, where
v approaches
c.
Second, while it's true that the limit of the field due to the
charge carriers in a short wire as the length approaches zero is the
same as the field of a point charge, we assumed in our use of equation (
2) that the motion of our point charge was
uniform.
That's only true for a straight wire! If the wire bends,
the charge carriers are actually accelerating. Again, as long as
the charge carrier velocities are much smaller than
c we
can largely ignore the changes to the field introduced by the
acceleration, but none the less the consequence is that, from this
derivation, we can't tell whether the BiotSavart law is correct for
arbitrarily kinked wires and highvelocity currents, or whether it's
only a lowspeed approximation for use with smoothly curved wires.
Page
created on 1/17/2007