Calculating Energy Stored in a Toroidal Coil

[Reshma]

Calculate the energy stored in a toroidal coil.

The magnetic field in a toroid is give by: B = \frac{\mu_0 n I}{2\pi r}

Energy stored is given by:
W = {1\over 2\mu_0}\int B^2 d\tau

W = {1\over 2\mu_0} \frac{\mu_0^2 n^2 I^2}{4\pi^2}\int {1\over r^2}d\tau

How do I determine the volume element for a toroid?

 

[Meir Achuz]

The toroid can usually be approximated by a solenoid of length 2\pi R,
with r constant inside the toroid. Then V=(2\pi L)*(\pi a^2), with B constant.