- 136

- 1

**1. Homework Statement**

The following problem from Griffiths is irritating me for a long time…

A toroidal coil has a rectangular cross section, with inner radius a, outer radius a+w, and height h. It carries a total of N tightly bound wound turns, and the current is increasing at a constant rate dI/dt=k.If w and h are both much less than a, find the electric field at a point z above the centre of the toroid.

[Griffiths gives hint: exploit the analogy between Faraday fields and magnetostatic fields.]

**2. Homework Equations**

Maxwell's equations!!!

**3. The Attempt at a Solution**

Here are the two ways I approached the problem.

1. For each turn of wire, a rectangular loop may be assumed. For dI/dt, the magnetic flux Φ (=∫B∙da ) is increasing. So, induced emf ∫E∙dl = ξ = (-dФ/dt ).

Using standard expression of B (=μNI/2πs) (in the circumferential direction) inside the loop, I got ---

ξ = [(μ(N^2)hk)/2π] ln[(a+w)/a]

I understand that this closed line integral is evaluated around the rectangular loop. But I need to evaluate e field at the top of the toroid axis…Is there any way out?

2. Griffiths approach: E can be evaluated from divE=0 and curl E= (-∂B/∂t) and E→ 0 at ∞. Well, I still do not know how to find E(z) specified. I tried with a trick: taking curl of (curl of E) in LHS and curl of B in RHS. So that the LHS reduces to laplacian of E and from that a Poisson like equation should follow. However, the RHS got zero!!!

So this is the case. Please help and note that I need to understand the mathematics in physical terms.