The ratio of magnetic field at the centre of a current carrying...

The formula to calculate the magnetic field at any axial point situated at a distance $x$ from the centre of a circular current carrying conductor of radius $r$ can be written as

$B = \dfrac{{\mu}_{0}I{r}^{2}}{2{\left({r}^{2} + {x}^{2}\right)}^{3 / 2}}$ . . . \left(1\right)$</p><p>Substitute$0$for$x$in equation (1) to calculate the magnetic field \left({B}_{c}\right)$ at the centre of the conductor.

$\begin{cases}{B}_{c} & = & \dfrac{{\mu}_{0}I{r}^{2}}{2{\left({r}^{2} + {0}^{2}\right)}^{3 / 2}} \\ & = & \dfrac{{\mu}_{0}I{r}^{2}}{2{r}^{3}} \\ & = & \dfrac{{\mu}_{0}I}{2r} . . . \left(2\right)\end{cases}$

Substitute $r$ for $x$ in equation (1) to calculate the magnetic field \left({B}_{r}\right)$at the given distance from the centre of the conductor.</p><p>$\begin{cases}{B}_{r} & = & \dfrac{{\mu}_{0}I{r}^{2}}{2{\left({r}^{2} + {r}^{2}\right)}^{3 / 2}} \\ & = & \dfrac{{\mu}_{0}I{r}^{2}}{4\sqrt{2}{r}^{3}} \\ & = & \dfrac{{\mu}_{0}I}{4\sqrt{2}r} . . . \left(3\right)\end{cases}$</p><p>Divide equation (2) by equation (3) to calculate the required ratio of the magnetic fields.</p><p>$\begin{cases}\dfrac{{B}_{c}}{{B}_{r}} & = & \dfrac{\dfrac{{\mu}_{0}I}{2r}}{\dfrac{{\mu}_{0}I}{4\sqrt{2}r}} \\ & = & 2\sqrt{2} \\ & = & \sqrt{8}\end{cases}