Question

The magnetic field normal to the plane of a wire of n turns and radius r which carries a current i is measured on the axis of the coil at a small distance h from the centre of the coil. This is smaller than the magnetic field at the centre by the fraction

• A. $\left(2/3\right)r^2/h^2$
• B. $\left(3/2\right)r^2/h^2$
• C. $\left(2/3\right)h^2/r^2$
• D. $\left(3/2\right)h^2/r^2$

Answer 1

The correct option is D $\left(3/2\right)h^2/r^2$

Magnetic field at the centre of the coil, $B_c=\frac{{\mu }_{o}ni}{2r}$

Magnetic field at an axial point , $B=\frac{{\mu }_{o}ni{r}^2}{2(r^2+h^2{)}^{\frac{3}{2}}}$

For $h<<r,B=\frac{{\mu }_{o}ni}{2r}[1-\frac{3}{2}\frac{h^2}{r^2}]$

Now fractional change$\delta =\frac{B_c-B}{B_c}$

$⟹\delta =\frac{3}{2}\frac{h^2}{r^2}$