question_answer

A ring of radius R having a linear charge density \[\lambda \] moves towards a solid imaginary sphere of radius \[\frac{R}{2},\] so that the centre of ring passes through the centre of sphere. The axis of the ring is perpendicular to the line joining the centres of the ring and the sphere. The maximum flux through the sphere in this process is:

A) \[\frac{\lambda R}{{{\in }_{0}}}\]

B) \[\frac{\lambda R}{2{{\in }_{0}}}\]

C) \[\frac{\lambda \pi R}{4{{\in }_{0}}}\]

D) \[\frac{\lambda \pi R}{3{{\in }_{0}}}\]

Correct Answer: D

Solution :

Flux will be maximum when maximum length of ring is inside the sphere.

This will occur when the chord AB is maximum. Now maximum length of chord AB= diameter of sphere. In this case the arc of ring inside the sphere subtends an angle of \[\frac{\pi }{3}\] at the centre of ring.

\[\therefore \]      charge on this \[arc=\frac{R\pi }{3}\cdot \lambda \]

\[\therefore \]      \[\phi =\frac{\frac{R\pi }{3}\lambda }{{{\in }_{0}}}=\frac{R\pi \lambda }{3{{\in }_{0}}}\]