Question

A thin wire of length $l$ is carrying a constant current. The wire is bent to form a circular coil. If the radius of the coil, thus formed, is equal to $R$ and the number of turns in it is equal to 𝑛, then which of the following graphs represent $\left(s\right)$ variation of magnetic field induction $\left(B\right)$ at the centre of the coil?

• A. Diagram
• B. Diagram
• C. Diagram
• D. Diagram

Answer 1

The correct option is D Diagram

Given,
Length of the wire is $l$
Radius of circular coil is $R$
No. of turns $n$
Then, total length of wire in loop
$\Rightarrow l=nx\left(2\pi R\right)$
$\Rightarrow n=\frac{1}{2\pi R}$

Magnetic field at centre of coil,

$B=\frac{{\mu }_{0}ni}{2R}=\frac{{\mu }_0}{4R}=\left(\frac{2\pi ni}{R}\right)=\frac{{\mu }_{0}li}{4\pi R^2}$

Where,$i$ is current in coil,
Then we can write,
$B\propto \frac{1}{R^2}$
So, $B$ is inversly poportional to $R^2$
Therefore, option $B$ and $D$ are wrong

Also,
$B=\frac{{\mu }_0}{4\pi }\left(\frac{2\pi ni}{R}\right)=\frac{{\mu }_{0}ni}{2\left(\frac{l}{2\pi n}\right)}=\frac{{\mu }_{0}i\pi n^2}{l}$
$\Rightarrow B\propto n^2$

This means the graph between $B$ and $n$ will be a parabola having increasing slope and passing through $(0,0)$.