Physics - Magnetism Question with Solution | TestHub

The magnetic field $(B)$ at a point induced by a flowing current $(I)$ is given as, $B=\frac{{\mu }_{0}I}{2r}$ Where ${\mu }_0$ is the permeability of free space, and $r$ is the radius of the circular path. Substituting for $I$ and rearranging, $\begin{array}{rl}& B=\frac{{\mu }_0\left(\frac{ev}{2\pi r}\right)}{2r}\\ \Rightarrow & B=\frac{{\mu }_{0}ev}{4\pi r^2}\\ \Rightarrow & r^2=\frac{{\mu }_{0}ev}{4\pi B}\\ \Rightarrow & r^2\propto \frac{v}{B}\\ \therefore & r\propto \sqrt{\frac{v}{B}}\end{array}$ Therefore, when an electron moves in a circular orbit with a uniform speed $v$ and produces a magnetic field $B$ at the centre of the circle, the radius of the circle is proportional to $\sqrt{\frac{v}{B}}$.