A solenoid `60cm` long and of radius `4*0cm` has 3 layers of windings of `300` turns each. A `2*0cm` long wire of mass `2*5g` lies inside the solenoid (near its centre) normal to the axis : both the wire and the axis of the solenoid are in the horizontal plane. The wire is connected through two leads parallel to the axis of solenoid to an external battery which supplies a current of `6*0A` in the wire. What value of current (with appropriate sense of circulation) in the windings of the solenoid can support the weight of the wire? `g=9*8ms^-2`.

Explanation

The wire lies in the magnetic field of solenoid. The force due to field balances the weight.

Here, number of layers of windings = 3
Number of turns in each layer = 300
Length of the solenoid ` = 60 cm = 0.6 m `
` :. ` Number of turns per unit length,
` n = (3 xx 300)/(0.6) = 1,500`
Let I be the current needed through the solenoid. The magnetic field inside the solenoid will be:
` B = mu_(0) nI`
The force acts on the wire normal to its length and is given by
` :. " " F = BI. l`
Here, ` I. = 6 A " and " l = 2 cm = 0.02 m`
If m is the mass of the wire and g is the acceleration due to gravity, then
`mu_(0)nII.l = mg`
Here , `m = 2.5 g = 2.5 xx 10^(-3) kg`
` :. " " I = (mg)/(mu_(0)n I.l) `
`rArr" " I = (2.5 xx 10^(-3) xx 9.8 xx 7)/(4 xx 22 xx 10^(-7) xx 1,500 xx 6 xx 0.02) `
` = 108.27 A`