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Under the influence of a uniform magnetic field a charged particle is moving in a circle of radius $R$ with constant speed $v$. The time period of the motion
depends on both $R$ and $v$
is independent of both $R$ and $v$
depends on $R$ and not on $v$
depends on $v$ and not on $R$
Solution
When a particle moves in a magnetic field of intensity B pointing downwards into the page and particle is moving with a speed of $v$ on the plane of paper:
$F = qvB$ (Force of charged particle in a magnetic field)
And we know that
$F =\frac{ mv ^2}{ r } \quad$ (r is the radius of motion and $m$ is mass of particle)
$\Rightarrow qvB =\frac{ mv ^2}{ r }$
$\Rightarrow r =\frac{ mv }{ Bq }$
Now as we know that
$\omega=\frac{ V }{ r }$
$\Rightarrow \omega=\frac{ Bq }{ m }$
Time period, $T =\frac{2 \pi}{\omega}$
$\Rightarrow T =\frac{2 \pi m }{ Bq }$
And this shows that it is independent of both radius and velocity.
