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.
