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

Here, for solenoid, `l=60cm=0*60m, N=3x300=900`:
For wire, `l_1=2.0cm=0*20m, m=2*5g=2*5xx10^-3kg, I_1=6*0A`
Let current I be passed through the solenoid windings, the magnetic field produced inside the solenoid due to current is `B=(mu_0NI)/(l)`
Force on wire `=I_1l_1B=I_1l_1(mu_0NI)/(l)`
The wire can be supported if the force on wire is equal to the weight of wire, i.e.,
`I_1l_1(mu_0NI)/(l)=mg` or `I=(mgl)/(I_1l_1mu_0N)=(2*5xx10^-3xx9*8xx0*6)/((6*0)xx(0*02)xx(4pixx10^-7)xx900)=108*27A`