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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.

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