A non - conducting sphere has mass of 100 g and radius 20 cm . A flat compact coil of wire with turns 5 is wrapped tightly around it with each turns concentric with the sphere. This sphere is placed on an inclined plane such that plane of coil is parallel to the inclined plane.
A uniform magnetic field of `0.5` T exists in the region in vertically upwards direction. Compute the current I required to rest the sphere in equilibrium.
` (##SUR_PHY_XII_V01_C03_E04_005_Q01##) `

Explanation

The sphere is in translational equilibrium , thus `f_(s) - mg sin theta ` = 0 ….(1)
The sphere is in rotational equilibrium . If torques are taken about the centre of the sphere, the magnetic field produces a clockwise torque of magnitude
i.e `tau = mB sin q [mu = NIA]`
The frictional force `(f_(s))` produces a anticlockwise torque of magnitude `tau = f_(s)R`, where R is the radius of the sphere. Thus
` f_(s) R - mB sin theta = 0 ` .....(2)
From (1) and (2) [i.e `f_(s) = mg sin theta ` substituting in (2)] mg `sin theta . R - mu B sin theta mg R = mu B `
Substituting ` mu `
mgR = NIAB
` I = "mgR"/"NBA" ` [where A is the area of the sphere ` A = pi R^(2) ` ]
` :. I = "mg"/(pi"RBN")`
Given :
mass of the sphere ` mu = 100 g = 100 xx 10^(-3) kg `
Radius of the sphere ` R = 20 cm = 20 xx 10 ^(-2) m `
No. of turns of wire wrapped ` N = 5`
Magnetic field ` B = 0.5 ` T
Current required to rest the sphere in equilibrium ` I = ? `
`I = (100 xx 10^(-3) xx cancel10^(2))/(pi xx cancel5 xx 20 xx 10^(-2) xx 0.5) `
` I = 2/pi A `