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?

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The cross-sectional area of the space to be filled is fixed, whilst that of the wire varies as [latex] {{d}^{2}} [/latex]. Thus n ∝ [latex] {{d}^{-2}} [/latex]. The resistance of one turn is inversely proportional to the cross-sectional area of the wire, i.e. varies as [latex] {{d}^{-2}} [/latex], and hence the resistance per unit length of the solenoid is R ∝ [latex] {{nd}^{-2}} [/latex] ∝ [latex] {d}^{-4} [/latex]. The flux density B is ∝ nI and therefore the required current I ∝ [latex] {{n}^{-1}} [/latex] ∝ [latex] {{d}^{2}} [/latex]. The heat dissipated per unit length is [latex] {{RI}^{2}} [/latex], which is ∝ [latex] {d}^{-4} [/latex] [latex] {{\left ( {{{d}^{2}}} \right )}^{2}} [/latex], 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.

Question : The turns of a solenoid, designed to provide a given magneti...

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