## Question:

The turns of a solenoid, designed to provide a given magnetic flux density along its axis, are wound to fill the space between two concentric cylinders of fixed radii. How should the diameter d of the wire used to be chosen so as to minimize the heat dissipated in the windings?

## Step-by-step

The cross-sectional area of the space to be filled is fixed, whilst that of the wire varies as ${{d}^{2}}$. Thus n ∝ ${{d}^{-2}}$. The resistance of one turn is inversely proportional to the cross-sectional area of the wire, i.e. varies as ${{d}^{-2}}$, and hence the resistance per unit length of the solenoid is R ∝ ${{nd}^{-2}}$${d}^{-4}$. The flux density B is ∝ nI and therefore the required current I ∝ ${{n}^{-1}}$${{d}^{2}}$. The heat dissipated per unit length is ${{RI}^{2}}$, which is ∝ ${d}^{-4}$ ${{\left ( {{{d}^{2}}} \right )}^{2}}$, i.e. independent of d. Thus (within limits) it does not matter what diameter wire is chosen so far as the heating effect is concerned.