Question

two small identical concentric circular coils marked 1 and 2 carry equal currents and are placed with their geometric axes perpendiculato each other derive an expression for the resultant magnetic field at o

Answer 1

$$\begin{gathered} \mathrm{Magnetic}\mathrm{field}\mathrm{due}\mathrm{to}\mathrm{each}\mathrm{circular}\mathrm{coil}: \\ \mathrm{B}=\frac{{\mathrm{\mu }}_0}{2}\frac{\mathrm{I}}{\mathrm{r}} \\ \mathrm{Net}\mathrm{magnetic}\mathrm{field}\mathrm{due}\mathrm{to}\mathrm{both}\mathrm{coils}: \\ \mathrm{As}\mathrm{both}\mathrm{coils}\mathrm{axes}\mathrm{are}\mathrm{perpendicular}\mathrm{to}\mathrm{each}\mathrm{other}, \\ {\mathrm{B}}_{\mathrm{net}}=\sqrt{{\mathrm{B}}^2+{\mathrm{B}}^2} \\ =\sqrt{2}\mathrm{B} \\ =\sqrt{2}\left[\frac{{\mathrm{\mu }}_0}{2}\frac{\mathrm{I}}{\mathrm{r}}\right] \\ \mathbf{}\mathbf{}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}}{\sqrt{\mathbf{2}}}\frac{\mathbf{I}}{\mathbf{r}} \end{gathered}$$