Introductory Physics Volume Two

108 Sources of Magnetic Field 5.4
distance r from an infinite wire was µ0I
2πy
. Let us take the limit of a → ∞
of the results we have for the finite wire, and see if we get the same
results.
⎛
⎞
µ 0 I x +
B = lim ⎝
a 2
x − a 2
√
a→∞ 4πy (x )
− √ ⎠
+
a 2
2 + y
2 (x −
a
2 )2 + y 2
(
= µ a
0I
2
√
4πy (
a
− − a 2
√
2 )2 (−
a
2 )2 )
= µ 0I
2πy
OK
Example
A circular wire with radius a carryies a current I. Locate the circle
in the xy-plane and compute the magnetic field it causes along a line
through its center.
Here is the geometry of the loop:
z
r f
r s
The path along the circle can be parameterized using t = [0, 2π]:
We also have
dl
⃗r s = a cos t î + a sin t ĵ
dr ⃗ s
= −a sin t î + a cos t ĵ
dt
⃗r f = z ˆk
⃗r f − ⃗r s = −a cos t î − a sin t ĵ + z ˆk
√
| ⃗r f − ⃗r s | = a 2 cos 2 t + a 2 sin 2 t + z 2
= √ a 2 + z 2
Compute the cross product needed for the Biot-Savart law:
⃗ dr s
dt × (⃗r f − ⃗r s ) = (−a sin tî + a cos tĵ) × (−a cos tî − a sin tĵ + zˆk)
= az cos tî + az sin tĵ + (a 2 sin 2 t + a 2 cos 2 t)ˆk
= az cos tî + az sin tĵ + a 2ˆk