A solenoid 60 cm long and of radius 4 cm has 3 layer of windings 300 turns each. A 2.3 cm long wire of mass 2.5g lies inside the solenoide near its center normal to its 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 the solenoid to an external battery which supplies a current of 6A 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?

Explanation

Length of the solenoid, `L = 60 cm = 0.6m`
Radius of the solenoid, `r = 4.0 cm = 0.04m`
It is given that there are 3 layer of windings of 300 turns each.
`:.` Total number of turns, `n = 3 xx 300 = 900`
Length of the wire, `l = 2cm = 0.02m`
Mass of the wire, `m = 2.5 g = 2.5 xx 10^(-3)kg`
Current flowing through the wire, `i = 6A`
Acceleration due to gravity, `g = 9.8 m//s^(2)`
Magnetic field product inside the solenoid, `B = (mu_(0)nI)/(L)`
where,
`mu_(0) =` Permebaility of free space `=4pi xx 10^(-7)TmA^(-1)`
I =Current flowing through the windings of the solenoid Magnetic force is given by the relation,
`F = Bil`
`=(mu_(0)nI)/(L) il`
Also, the force on the wire is equal to the weight of the wire.
`:. mg = (mu_(0)nlil)/(L)`
`I = (mgL)/(mu_(0)nil)`
`= (2.5 xx 10^(-3) xx 9.8 xx 0.6)/(4pi xx 10^(-7) xx 900 xx 0.02 xx 6) = 108A`
Hence, the current flowing through the solenoid is 108A.