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*From*: Leigh Palmer <palmer@SFU.CA>*Date*: Thu, 1 Feb 2001 09:56:39 -0800

A related question:

Is it a simple matter to calculate the electric field in the region

between a circular cylinder of charge Q1 and an elliptical cylinder

of charge Q2? Assume the circular cylinder has a radius a < b (the

semi-minor axis of the elliptical cylinder), they share an axis of

symmetry, and the region in between is a vacuum.

The outer cylinder must be nonconducting to meet your specification.

If it is a conductor, Q2 = -Q1. (I assume the cylinders are long.)

You should also specify the length and semimajor axis, and specify a

constraint on the charge distribution (e.g. uniform surface charge

density on the elliptical shell). Simple versions of this problem are

solved simply using the principle of superposition: the field due to

the two cylinders is just the (vector) sum of the fields due to the

individual cylinders. They don't interact.

Since your subject line refers to shielding I will assume that you are

thinking of the case in which both cylinders are conducting.

I know that if the outer cylinder is also circular the electric

field in between is a function of Q1 and r only, but I suppose that

if the outer cylinder is not circular this is no longer true. Is

this correct?

Justin Parke

Oakland Mills High School

Dear Justin,

You are correct; it is no longer true. I'm very rusty on what problems

are capable of solution in closed form, but if the one you propose is

one of those, it is too hard for me to solve, and the mathematics may

be over your head, too. Failing a closed form solution, let me suggest

a couple of approaches you might follow.

Lines of force (or electric field) must intersect the surface of a

conductor normally (perpendicular to the surface) if no currents are

flowing. The number of lines of force per unit area is proportional to

the electric field strength. The electric field is strongest along the

semiminor axis of the ellipse. You should draw what you think the

field might look like if it had to conform to these conditions. That

drawing should help you answer your questions.

Although the mathematics of this problem is difficult, it is amenable

to approximate solution by numerical means. A relaxation calculation

is easy and fun using a spreadsheet (like Excel) and one of the

improbably powerful computers that seem to be so common today. In this

case the geometry is difficult, but not impossible. You would be well

advised, however, to start with a simpler problem, like the electric

field between concentric rectangular cross-section conductors, better

suited to the rectangular data structure of the spreadsheet.

I think you would learn a lot from this exercise, and it would warm my

heart to see these game machines (that's what is making supercomputers

popular) turned to nobler uses. I have ordered a laptop that is so

powerful that I would be breaking the law if I took it with me to

Russia! We live in amazing times.

Leigh

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