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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 :
A
$\sqrt{\dfrac{v}{B}}$
B
$\dfrac{v}{B}$
C
$\sqrt{\dfrac{B}{v}}$
D
$\dfrac{B}{v}$
Detailed Solution
The revolving electron is equivalent to a current $I = \dfrac{e}{T}$, where the time period is $T = \dfrac{2\pi R}{v}$.
$I = \dfrac{ev}{2\pi R}$
Magnetic field at the centre of a circular current: $B = \dfrac{\mu_0 I}{2R}$
$B = \dfrac{\mu_0}{2R}\left(\dfrac{ev}{2\pi R}\right) = \dfrac{\mu_0 ev}{4\pi R^2}$
$\Rightarrow R^2 = \dfrac{\mu_0 ev}{4\pi B}$
$\Rightarrow R \propto \sqrt{\dfrac{v}{B}}$
$I = \dfrac{ev}{2\pi R}$
Magnetic field at the centre of a circular current: $B = \dfrac{\mu_0 I}{2R}$
$B = \dfrac{\mu_0}{2R}\left(\dfrac{ev}{2\pi R}\right) = \dfrac{\mu_0 ev}{4\pi R^2}$
$\Rightarrow R^2 = \dfrac{\mu_0 ev}{4\pi B}$
$\Rightarrow R \propto \sqrt{\dfrac{v}{B}}$