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Q.
An electron moves in a circular orbit with a uniform speed $ v $ . It produces a magnetic field $B$ at the centre of the circle. The radius of the circle is proportional to
The time period of electron moving in a circular orbit
$T=\frac{\text { circumference of circular path }}{\text { speed }}$
$=\frac{2 \pi r}{v}$
and equivalent current due to electron flow
$I=\frac{e}{T}=\frac{e}{(2 \pi r / v)}=\frac{e v}{2 \pi r}$
Magnetic field at centre of circle
$ B=\frac{\mu_{0} I}{2 r}=\frac{\mu_{0} e v}{4 \pi r^{2}} $
$\Rightarrow r \propto \sqrt{\frac{v}{B}}$