W9_parallel_resonance.eps

234 Week 7: Sources of the Magnetic Field

A

dl

r

q

θ

Ι

vd

dB (in)

Figure 79: The microscopic model of conduction used to derive the Biot-Savart Law from the
magnetic field of a point charge.

origin of~r, at the point~ris:

B~=μ^0

4 π

q(~vd×rˆ)

r^2 (501)

If we take a very small chunk of wire (and remember, we are exaggerating its width, which is really
small relative tor) of lengthdℓthenallof the charges in the volumeA dℓare moving with velocity~vd
in the direction of the current, and have approximately the same vector~rto the point of observation.
The total field ofallof these charges, in any chunk that contains enough charges so that the idea of
the density of charge carriers makes sense, is produced by the total charge in this volume.

That charge is:

dQ=nqAdℓ (502)

and the field it produces is thus:

dB~ =

μ 0

4 π

dQ(~vd׈r)

r^2

=

μ 0

4 π

nqAdℓ(~vd×rˆ)

r^2

(503)

We now play a clever trick. We make avectord~ℓ=dℓˆvdthat points in the same direction as~vd,
leaving behind its magnitudevd, so that this equation becomes:

dB~=μ^0

4 π

nqAvd(d~ℓ׈r)

r^2

(504)

and identifyI=nqAvdfrom our previous conduction model! This then becomes:

dB~=km

I(d~ℓ×rˆ)

r^2

(505)

which is theBiot-Savart(Bee-oh Sah-vahr) Law!

We will usually write this more properly in more general coordinates instead of coordinates that
centered (as these ones did) on the chunkd~ℓ, and for a possibly curved conductor instead of just a
straight one. The result (and its associated figure 80 becomes:

dB~=km

I(d~ℓ×(~r−~r 0 ))

|r−r 0 |^3

(506)