Question

A wire in the form of a circular loop of one turn carrying a current produces a magnetic field $B$ at the centre. If the same wire is looped into a coil of two turns and carries the same current, the new value of magnetic induction at the centre is:

• A. $3B$
• B. $5B$
• C. $4B$
• D. $2B$

Answer 1

Answer: (C)

$4B$

Let the radius of the circular loop be $r$

Magnetic field at its centre $B=\frac{{\mu }_o}{2\pi }\frac{I}{r}$ .........(1)

Thus length of the wire $L=2\pi r$

But the same wire is looped into a coil of radius $r_2$ of two turns.

$\therefore$ $L=2\left(2\pi r_2\right)=2\pi r$ $⟹r_2=\frac{r}{2}$

Magnetic field at its centre $B_2=\frac{{\mu }_o}{2\pi }\frac{I}{r_2}\times n$

$\therefore$ $B_2=\frac{{\mu }_o}{2\pi }\frac{I}{r/2}\times 2=4\times \frac{{\mu }_{o}I}{2\pi r}$

From (1) we get $B_2=4B$