A uniform magnetic field B is present in a region. An electron is...

When an electron moves in a circular path perpendicular to a uniform magnetic field $B$, the magnetic force provides the necessary centripetal force:

$e v B = \dfrac{m v^2}{r}$

$r = \dfrac{m v}{e B}$

According to Bohr's quantization rule for angular momentum:

$m v r = \dfrac{n h}{2 \pi}$

Substituting the value of $r$ in the above equation:

$m v \left( \dfrac{m v}{e B} \right) = \dfrac{n h}{2 \pi}$

$\dfrac{m^2 v^2}{e B} = \dfrac{n h}{2 \pi}$

$v^2 = \dfrac{n h e B}{2 \pi m^2}$

$v = \sqrt{\dfrac{n h e B}{2 \pi m^2}}$

For the minimum possible speed, we take the minimum principal quantum number, $n = 1$:

v_{\text\min} = \sqrt{\dfrac{h e B}{2 \pi m^2}}