An electron moves in a circular orbit with a uniform speed v . It...

The magnetic field $(B)$ at a point induced by a flowing current $(I)$ is given as, (B=\frac{\mu_0 I}{2 r}) Where (\mu_0) is the permeability of free space, and $r$ is the radius of the circular path. Substituting for $I$ and rearranging, ($$\begin{aligned} & B=\frac{\mu_0\left(\frac{e v}{2 \pi r}\right)}{2 r} \\ \Rightarrow & B=\frac{\mu_0 e v}{4 \pi r^2} \\ \Rightarrow & r^2=\frac{\mu_0 e v}{4 \pi B} \\ \Rightarrow & r^2 \propto \frac{v}{B} \\ \therefore & r \propto \sqrt{\frac{v}{B}} \end{aligned}$$) 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}}).