Question

Due to the flow of current in a circular loop of radius R, the magnetic field produced at the centre of the loop is B. The magnetic moment of the loop is:
A. $\frac{B{{R}^{3}}}{2{{\pi }^{3}}{{\mu }_{0}}}$
B. $\frac{2\pi B{{R}^{3}}}{{{\mu }_{0}}}$
C. $\frac{B{{R}^{2}}}{2{{\pi }^{2}}{{\mu }_{0}}}$
D. $\frac{2\pi B{{R}^{2}}}{{{\mu }_{0}}}$

Answer 1

Hint:We know that when a current flows through a loop then there will be a magnetic field associated with it. Here there is a circular loop with current flowing through it. We have to find the magnetic moment of the loop due to the magnetic field produced.



Formula used:

We have the equation for magnetic field due to circular loop at the centre is:
$B=\frac{{{\mu }_{0}}i}{2R}$
Where ${{\mu }_{0}}$is the permeability of free space, i is the current flowing through the loop and R is the radius of the loop.
And we have equation for magnetic moment as:
$M=i\times A$
Where is the current flowing through the loop and A is the area of the loop.

Complete answer:
We have a circular loop of radius R through which a current of magnitude i is flowing. Due to this current flowing through the loop magnetic field of magnitude B is produced at the centre of loop.
We have to find the magnetic moment of the loop.
First of all, we have equation for magnetic field at the centre of loop as:
$B=\frac{{{\mu }_{0}}i}{2R}$
From which we can formulate an equation for current flowing through the circular loop in terms of magnetic field.
That is,
Current flowing through the loop, $i=\frac{B\times 2R}{{{\mu }_{0}}}$
Now we have equation for magnetic moment as:
$M=i\times A$
In this question we are considering circular loop so area of the loop will be
$A=\pi {{R}^{2}}$
On substituting values of area and current in the equation of magnetic loop we get:
Magnetic moment of loop, $M=\frac{B\times 2R\times \pi {{R}^{2}}}{{{\mu }_{0}}}=\frac{2\pi B{{R}^{3}}}{{{\mu }_{0}}}$
Therefore, the answer is option (B)




Note:Remember that in this question we deal with circular loop and magnetic field produced at the centre of loop. If shape changes, then formula changes accordingly.