How should the diameter of the wire used be chosen so to minimize the heat dissipated in windings ? Check the details in question.

The turns of a solenoid is designed to provide a given magnetic flux density along its axis are wound to fill the space between 2 concentric cylinder of fixed radii.How should the diameter of the wire used be chosen so to minimize the heat dissipated in windings ?

Mar 7, 2018

This is what I get.

Explanation:

We are required to fill the cross sectional area between between two concentric cylinders of fixed radii with wire of diameter $d$.

$\implies$ The cross sectional area to be filled is fixed.
Cross section of wire $= \frac{\pi {d}^{2}}{4}$
$\implies$ Cross section of wire $\propto {d}^{2}$
$\therefore$Number of turns per unit length $n \propto {d}^{-} 2$ ........(1)

Resistance of one turn of wire $\propto {\text{Area of cross section of wire}}^{-} 1$
$\implies$ Resistance of per unit length of wire $R \propto n {d}^{-} 2$

$R \propto {d}^{-} 4$ .......(2)

We know that magnetic flux density produced $B = \mu I n$
$\implies$ For the fixed magnetic field $I \propto {n}^{-} 1$
From (1) above becomes

$I \propto {d}^{2}$ .....(3)

Heat dissipated in a resistance $= {I}^{2} R$
Using (2) and (3)
Heat dissipated in a resistance $\propto {\left({d}^{2}\right)}^{2} \left({d}^{-} 4\right)$

$\implies$Heat dissipated in a multilayered solenoid, as described above, is independent of diameter of wire.