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 ?

1 Answer
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 dd.

=> The cross sectional area to be filled is fixed.
Cross section of wire =(pid^2)/4=πd24
=> Cross section of wire propd^2d2
:.Number of turns per unit length npropd^-2 ........(1)

Resistance of one turn of wire prop"Area of cross section of wire"^-1
=> Resistance of per unit length of wire Rpropnd^-2

Rpropd^-4 .......(2)

We know that magnetic flux density produced B=muIn
=> For the fixed magnetic field Ipropn^-1
From (1) above becomes

Ipropd^2 .....(3)

Heat dissipated in a resistance =I^2R
Using (2) and (3)
Heat dissipated in a resistance prop(d^2)^2(d^-4)

=>Heat dissipated in a multilayered solenoid, as described above, is independent of diameter of wire.