W9_parallel_resonance.eps

246 Week 7: Sources of the Magnetic Field

(infinite)

b

b

I (in) C

B

Figure 88: A cross-sectional view of an infinitely long solenoid withnturns per unit length, cross-
sectional areaA, carrying currentIin each turn. The field both inside and outside of the solenoid
is parallel to the axis of the solenoid (from symmetry), leading to theAmperian Path shown.

If you examine figure 88, you can see from symmetry that the magnetic field inside must travel
parallel to the axis of the solenoid from right to left. The general right to left direction follows from
the right hand rule given the current into the page on the tops of allof the wires and out at the
bottom. The fact that it must beparallefollows from the fact that every point is in the middle
of an infinite line, so there can be no up or down or in or out compontent because it wouldn’t be
symmetric with respect to either inversion or translation down the solenoid to another “central”
point. Furthermore, the field strength must beconstantalong any straight line parallel to the axis
for the same reason – it cannot vary from its value in “the middle”, whereever you choose to put
that middle.

Outside the same is true but opposite. The field (if any) must flow from left to right and be
parallel to the axis of the solenoid. This determines a good Amperian PathC. We select a rectangle
of sideb(inside the solenoid) withinfinitely long sides! The field is everywhere perpendicular to
the sides so we getno contributionto the path integral of the field from them. By making the sides
infinite, we can also make the field zero on the upper horizontal chunk. We only get a contribution
from the side of lengthbinside the solenoid. That is:
∮

C

B~·d~ℓ = Bzb+ 0(left) + 0(top) + 0(right) =μ 0 Ithru C

Bzb = μ 0 nbI

Bz = μ 0 nI=

μ 0 N I

L (541)

where we computed the total currentthroughCby multiplying the number of turns per unit length
by the length ofCthrough which the turns passed times their current.

Note well that this tells us that the field iszerooutside of an ideal solenoid – all magnetic field
lines are confined to live inside the solenoid tube and none can escape to the outside. It also tells us
that the field inside is uniform – there is no dependence of the answeron any spatial coordinates,
so it doesn’t vary with coordinates beyond being non-zero on the inside and zero on the outside.

The final form is given as you might use it for a solenoid with a finite number of turnsNand of
finite lengthL, where (recall)Lneeds to be much larger than the radius or diameter of the solenoid
and where we are finding the field not too near the ends. Usually we willidealize even finite size
solenoids as having the field of an infinite solenoid inside, and will neglectend effects. That is, we
will assume that the field is uniform but drops to zero “instantly” at the solenoid ends. Of course
this isn’t physical, but the fielddoesdrop off very rapidly at the ends, so it is a good approximation