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
A08
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#.

#=># The cross sectional area to be filled is fixed.
Cross section of wire #=(pid^2)/4#
#=># Cross section of wire #propd^2#
#:.#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.

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