W9_parallel_resonance.eps

Week 7: Sources of the Magnetic Field 245

Then, region II (r′> R):

∮′

C

B~·d~ℓ = μ 0

∫

S/C

J~·nˆdA

Bt 2 πr′ = μ 0 I

Bt =

μ 0 I

2 πr′

(540)

which is the same as for a long straight thin wire. The fieldoutsideof any cylindrical current will be
the same as the field of a current of the same strength all concentrated in a thin wire at the origin.
This should all be very reminiscent of Gauss’s Law and fields outside ofcylinders or spheres.

B

R r

0

2 Rπ

μI

Figure 87:B(r) for a long thick wire of radiusRcarrying a currentI. Note that the field increases
linearly inside of the wire and reaches a maximum value on the surface of the wire. Outside it drops
off like 1/r. Although the field is continuous, its derivative (slope) is not; it jumps atr=R.

We crudely plot the field as a function ofrin figure 87. Remember, the field circulates around
the current (density) in a clockwise direction as determined by the right hand rule.

We could, of course, do more complicated problems now that have this symmetry as long as we
can figure out how to do the integrals (or otherwise figure out the amount of current that passes
throughC) on therighthand side of Ampere’s Law. The left hand side is always the same. Variations
include: Finding the field in a thick cylindrical shell carrying a currentI; a coaxial cable; a thick
wire with a cylindrical hole, a thick wire with a current density that isnotuniform. The latter is
particularly relevant for alternating currents – when an alternating current is sent through a thick
wire the current isnotuniformly distributed, it tends to concentrate near the surface and die off
in the middle. This has implications for computing the resistance and actually affects the design of
high voltage power transmission lines and wave guides.

Example 7.6.3: The Solenoid

The solenoid pictured above in figure 88 is a classic problem in magnetism– it is (as we will see)
the moral equivalent of a capacitor for the storing ofmagneticenergy. A solenoid is also our ideal
model for “permanent magnets” as well as electromagnets of all flavors.

In order to apply Ampere’s Law to a solenoid – which is basically a cylindrical coil of wire with
many (N) turns and cross-sectional areaAcarrying a currentI– we need the solenoid to have
enoughsymmetrythat we can figure out a suitable Amperian Path. To accomplish this, we will
assume that the solenoid istightly wrapped– so much so that the coils form a more or lesscontinuous
current around the interior volume – and that it isinfinitely long. Both are idealizations, but both
of these assumptions aregoodidealizations – they will work well enough for any snugly wrapped coil
that is (much) longer than its diameter.