A solid, non-conducting sphere has charge $Q$ uniformly distributed over its surface and is rotated uniformly with a constant angular velocity $ω$ about its diameter (which coincides with z axis) as shown. The value of the integral $\int_{- \infty}^{+ \infty} \vec{B} \cdot \vec{d l}$ along the entire z axis (where B is due to moving charges on the sphere) is of magnitude $\frac{μ_{0} Q ω}{n π}$ . Find the value of $n$ .

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

answer is 2.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Imagine the infinite line as a closed loop with a small part of this infinite loop, having length $2 R$ , passing through the centre of the sphere. As the sphere is rotating, current threading the loop is equal to the charge on the sphere divided by time period of rotation

$i = \frac{Q}{T} = \frac{Q ω}{2 π}$

Now using Ampere's circuital law, for the closed loop,

$\oint \vec{B} \cdot \vec{d l} = μ_{0} i = \frac{μ_{0} Q ω}{2 π}$