Question

The fractional change in the magnetic field intensity at a distance ' $r$ ' from centre on the axis of current carrying coil of radius '$a$' to the magnetic field intensity at the centre of the same coil is: (Take $r<a$ )

• A. $\frac{3}{2}\frac{a^2}{r^2}$
• B. $\frac{2}{3}\frac{a^2}{r^2}$
• C. $\frac{2}{3}\frac{r^2}{a^2}$
• D. $\frac{3}{2}\frac{r^2}{a^2}$

Answer 1

Answer: (D)

$B_{\text{axis }}=\frac{{\mu }_{0}i{R}^2}{2{\left(R^2+x^2\right)}^{3/2}}$
$B_{\text{centre }}=\frac{{\mu }_{0}i}{2R}$
$\therefore B_{\text{centre }}=\frac{{\mu }_{0}i}{2a}$
$\therefore B_{\text{axis }}=\frac{{\mu }_{0}i{a}^2}{2{\left(a^2+r^2\right)}^{3/2}}$
$\therefore$ fractional change in magnetic field =
$\frac{\frac{{\mu }_{0}i}{2a}-\frac{{\mu }_{0}i{a}^2}{2{\left(a^2+r^2\right)}^{3/2}}}{\frac{{\mu }_{0}i}{2a}}=1-\frac{1}{{\left[1+\left(\frac{r^2}{a^2}\right)\right]}^{3/2}}$
$\approx 1-\left[1-\frac{3}{2}\frac{r^2}{a^2}\right]=\frac{3}{2}\frac{r^2}{a^2}$
Note : ${\left(1+\frac{r^2}{a^2}\right)}^{-3/2}\approx \left(1-\frac{3}{2}\frac{r^2}{a^2}\right)$
[True only if $r<<a$ ]