A circular coil of radius ' $r$ ' and number of turns ' $n$ ' carries a current ' $I$ '. The magnetic fields…
A circular coil of radius ' $r$ ' and number of turns ' $n$ ' carries a current ' $I$ '. The magnetic fields at a small distance ' $h$ ' along the axis of the coil $\left(B_a\right)$ and at the centre of the coil $\left(\mathrm{B}_{\mathrm{c}}\right)$ are measured. The ralation between $B_c$ and $B_a$ is
Magnetic field along the axis of the coil is:
$\mathrm{B}_{\mathrm{a}}=\frac{\mu_0}{4 \pi}\left[\frac{2 \pi \mathrm{nIr} \mathrm{r}^2}{\left(\mathrm{r}^2+\mathrm{h}^2\right)^{\frac{3}{2}}}\right]$
Magnetic field at the centre of the coil is:
$\begin{aligned}
\mathrm{B}_{\mathrm{C}} & =\frac{\mu_0}{4 \pi}\left(\frac{2 \pi \mathrm{nI}}{\mathrm{r}}\right) \\
\frac{\mathrm{B}_{\mathrm{C}}}{\mathrm{B}_{\mathrm{a}}} & =\frac{\left(\mathrm{r}^2+\mathrm{h}^2\right)^{\frac{3}{2}}}{\mathrm{r}^3} \\
\therefore \quad \mathrm{B}_{\mathrm{C}} & =\mathrm{B}_{\mathrm{a}}\left[1+\frac{\mathrm{h}^2}{\mathrm{r}^2}\right]^{\frac{3}{2}}
\end{aligned}$