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 be chosen so as to minimize the heat dissipated in the windings?

• Wire should be multiple of 5d

• Wire should be multiple of d/3

• the wire is independent of d

• Can't say

C.

the wire is independent of d

The cross-sectional area of the space to be filled is fixed. whilst, that of the wire varies as d². 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 ∝ d². The heat dissipated per unit length is RI², which ∝ d^-4(d²)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.