What is the magnetic field on the axis of a coil of radius r carrying current I at a distance R from the origin?

• $\frac{{\mathrm{\mu }}_0 \mathrm{I} {\mathrm{r}}^2}{2 {\left({\mathrm{r}}^2 + {\mathrm{R}}^2\right)}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$

• $\frac{{\mathrm{\mu }}_0 {\mathrm{IR}}^2}{2 {\left({\mathrm{r}}^2 + {\mathrm{R}}^2\right)}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$

• $\frac{{\mathrm{\mu }}_0\mathrm{I}}{2 \left(\mathrm{r} + \mathrm{R}\right)}$

• $\frac{{\mathrm{\mu }}_0 \mathrm{IR}}{2 {\left({\mathrm{r}}^2 + {\mathrm{R}}^2\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$

A.

$\frac{{\mathrm{\mu }}_0 \mathrm{I} {\mathrm{r}}^2}{2 {\left({\mathrm{r}}^2 + {\mathrm{R}}^2\right)}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$

Consider a circular coil of radius r, carrying a current I. Consider a point P, which is at a distance X from the centre of the coil. We can consider that the loop is made up of a large number of short elements, generating small magnetic fields. So the total field at P will be the sum of the contributions from all these elements. At the centre of the coil the field will be uniform. As the location of the point increases from the centre of the coil, the field decreases.

By Biot-Savart's law, the field dB due to a small element 'dl' of the circle, centred at A is given by

         dB = $\frac{{\mathrm{\mu }}_0 \mathrm{I}}{2} \frac{{\mathrm{r}}^2}{{\displaystyle {\left( {\mathrm{x}}^2 + {\mathrm{r}}^2 \right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$

This can be resolved into two components, one along the axis OP, ad other PS, which is perpendicular to OP.  Ps get cancelled with PS'. So the magnetic field at a distance x away from the axis of circular coil of radius r is given by

        B_x = $\frac{{\mathrm{\mu }}_0 \mathrm{n} \mathrm{I}}{2} \frac{{\mathrm{r}}^2}{{\displaystyle {\left({\mathrm{x}}^2 + {\mathrm{r}}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$