Question

A thin ring of radius R meter has charge q coulomb uniformly spread on it. The ring rotates about its axis with a constant frequency of f revolutions/s. The value of magnetic induction in Wb/m$^{2}$ at the centre of the ring is
A.)$\dfrac{{{\mu }_{0}}qf}{2\pi R}$
B.)$\dfrac{{{\mu }_{0}}q}{2\pi fR}$
C.)$\dfrac{{{\mu }_{0}}q}{2fR}$
D.)$\dfrac{{{\mu }_{0}}qf}{2R}$

Answer 1

Hint: We will first draw the diagram of the thin ring as per the given conditions and then , apply the formula of current in the circular conductor , and relate it with the frequency and current . We know that frequency is 1/t(time). Now as the value of magnetic induction is the one we have to know, hence Biot savart law will work in this part, replace the current in biot savart law with the derived value of i that will give the final result.

Complete step by step answer:

Here in the above diagram we see a ring with radius R and charge q coulomb,
Therefore current,

i=q/t, where ‘q’ is the charge and ‘t’ is the time period and ‘i’ is the current,
Now, in the question it is mentioned that the frequency is f, which means the ring has taken ‘f’ revolution in a time period ‘1 sec’,

Therefore, the current must be,

i=q/t=fq

where,

f is the frequency per second
q is the charge.

Now, we know that from Biot-savart Law,

That the current carrying circular conductor, had a magnetic field at its center that is represented as,
$B=\dfrac{{{\mu }_{\circ }}i}{2R}$ ,
Where i is the current, R is the radius and ${{\mu }_{\circ }}$ is the constant known as permeability of free space.
Now we know the value of ‘i’ is i=q/t=fq,

On replacing this value with the Biot savart law,

$B=\dfrac{{{\mu }_{\circ }}fq}{2R}$
Therefore, option D is the correct option.

Note: Representation of the question as a diagram must be correct, some may confuse between the relation of frequency and time period(Frequency is inversely proportional to time period), during replacing the current value in Biot Savart law one must be careful to represent it in the required form else the answer will not match.