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?

Step-by-step solution

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 n d⁻² d⁻⁴ . The flux density B is n I and therefore the required current I n⁻¹ d^2 . The heat dissipated per unit length is R I^2 , which d⁻⁴ (d^2 )^2 , i.e. independent of d . Thus, within limits, it does not matter what diameter