Question

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

• A. a quarter of its first value
• B. unaltered
• C. four times of its first value
• D. half of its first value

Answer 1

Answer: (C)

four times of its first value

Let the radii be $r_1$ and $r_2$ respectively.
Since there are two turns of radius $r_2$, $r_1=2r_2$
Magnetic field B at the center of the coil of radius $r_1$ $B_1=\frac{{\mu }_{o}i}{2r_1}=\frac{{\mu }_{o}i}{4r_2}$
Magnetic field B at the center of the coil of radius $r_2$ $B_2=2\times \frac{{\mu }_{o}i}{2r_2}$
$\therefore \frac{B_2}{B_1}=\frac{2\times \frac{{\mu }_{o}i}{2r_2}}{\frac{{\mu }_{o}i}{4r_2}}=4$