Question

A length $l$ of wire carries a steady current $i.$ It is bent first to form a circular plane coil of one turn. The same length is now bent more sharply to give three loops of smaller radius. The magnetic field at the centre caused by the same current is

• A. One-third of its first value
• B. Unaltered
• C. Three times of its initial value
• D. Nine times of its initial value

Answer 1

Answer: (D)

The magnetic field at the centre of a circular coil is
$B_{c e n t r e}=\frac{\mu _{0} \, i \, \times n}{2 r}$
Where $2\pi r=l= \, $ length of wire and n is number of turns. It is given that $2\pi r_{1}=l=3\left(2 \pi r_{2}\right)\Rightarrow r_{2}=\frac{r_{1}}{3}$
So $B_{1}$ at centre $=\frac{\mu _{0} i \, \times 1}{2 r_{1}}$
$B_{2}$ at centre $=\frac{\mu _{0} i \, \, \times 3}{2 r_{2}}$
So $\frac{B_{2}}{B_{1}}=\frac{3}{2 r_{2}}\times \frac{2 r_{1}}{1}=3\frac{r_{1}}{r_{2}}=3\frac{r_{1}}{\frac{r_{1}}{3}}=9$