Question

A circular loop of radius r, having n turns and carrying current i, is kept in the xy plane. It is then subjected to a uniform magnetic field b = bx i + by j + bz k. Obtain expression for the magnetic potential energy of the coil-magnetic field system.

Answer 1

The magnetic moment \vec{M}= N\vec{A}\iota

where A is cross sectional area of loop

A = πr² along z direction because loop placed in XY plane.

\vec{M}=N(\pi r^2)\iota\hat{k}

potential energy of the coil magnetic field system is P.E=\vec{M}.\vec{B}

P.E = N(\pi r^2)\iota\hat{k}.(B_x\hat{i}+B_y\hat{j}+B_z\hat{k})

P.E = \pi r^2N\iota B_z