Question

Consider a toroid of the circular cross-section of radius b, major radius R much greater than minor radius b. Find the total energy stored in the toroid. (I is current, N is a total number of turns)

Answer 1

Given, the toroid of the circular cross-section of radius $=b$

Major radius$=R$

Minor radius$=b$

Total energy stored in the toroid$=?$

When current$=I$

Total number of turus$=N$

The formula for the energy stored in a magnetic field

$E=\frac{1}{2}L{I}^2$

where $L=\frac{\mu N^{2}A}{I}=\frac{\mu N^{2}A}{I}=\frac{\mu N^2\times \pi b^2}{I}$

$B=\mu NI/L$

$\therefore E=\frac{1}{2}L{I}^2=\frac{1}{2}\times \frac{\mu N^2\times \pi b^2}{I}\times I^2=\frac{1}{2}\mu N^2\times \pi b^2\times I$