W9_parallel_resonance.eps

244 Week 7: Sources of the Magnetic Field

Example 7.6.2: Cylindrical Current Density – Field of an Infinitely Long Thick Wire

B

r

r’

B

J (in)

C

C’

R

Figure 86: An infinitely long thick wire of circular radiusRcarries a currentIinto the page as
drawn. We would like to find the magnetic field in all space.

Suppose we have an infinitely long straight wire that has some finite radiusRand is carrying a
currentIthat is uniformly distributed across the wire cross-section as shown in figure 86. We would
like to compute the magnetic field everywhere in space, both inside and outside of the wire.

Our first step is to transformIinto a current densityJ~into the page:

J~= I

πR^2

ˆz (538)

where thez-axis is into the page.

Next, we have to think just a bit about the field we expect to get. This step isessential– most
people who get this problem wrong get it wrong because they omit it, they haven’t thought about
the problem enough. The current has cylindrical symmetry, so thefield will too. We expect the field
lines to run in circles of constant magnitude around the center of symmetry in the middle of the
wire, in the clockwise direction as drawn from the right hand rule. Butwe donotexpect them to
have the same form inside the wire and outside of the wire. We therefore have to drawtwoAmperian
Paths, one (C) of radiusrin region Ir < R, the other (C′) of radiusr′in region IIr′> R. We
have to apply Ampere’s Lawtwice, once in each region.

Let’s do region I (r < R):

∮

C

B~·d~ℓ = μ 0

∫

S/C

J~·ˆndA

Bt

∮

C

dℓ = μ 0 J

∫

S/C

dA

Bt 2 πr = μ 0 Jπr^2

Bt =

μ 0 Jπr^2

2 πr

=

μ 0 Ir

2 πR^2

(539)

where we have selected a right-handed normalnˆinto the page so that the dot product ofJ~andnˆ
is just the magnitudeJ. The right hand side, as you can see, computes the total currentthat flows
throughthe curveC(inside the radiusr)! The left hand side is identical to what it was for the thin
wire (and what it will be for all other cylindrical problems).