A particle of charge $- q$ and mass $m$ moves in a circular orbit of radius $r$ about a fixed charge $+ Q$ . The relation between the radius of the orbit $r$ and the time period $T$ is:

Question

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Answer 1

The centrifugal force acting on charge $q$ is:
$F = \frac{k q Q}{r ^{2}}$
$\Rightarrow \frac{m v ^{2}}{r} = \frac{k q Q}{r ^{2}}$
$\Rightarrow V = \frac{k q Q}{r m}$
Now, $V = \frac{2 π r}{T}$
$\Rightarrow \frac{2 π r}{T} = \frac{q Q}{4 π ε _{0} r m}$
$\Rightarrow \frac{4 π ^{2} r ^{2}}{T ^{2}} = \frac{q Q}{4 π ε _{0} r m}$
$\Rightarrow r^{3} = \frac{q Q T ^{2}}{1 6 π ^{3} ε _{0} m}$

Answer 2

The centrifugal force acting on charge

q

is:

F = \frac{k q Q}{r ^{2}}

\Rightarrow \frac{m v ^{2}}{r} = \frac{k q Q}{r ^{2}}

\Rightarrow V = \frac{k q Q}{r m}

Now,

V = \frac{2 π r}{T}

\Rightarrow \frac{2 π r}{T} = \frac{q Q}{4 π ε _{0} r m}

\Rightarrow \frac{4 π ^{2} r ^{2}}{T ^{2}} = \frac{q Q}{4 π ε _{0} r m}

\Rightarrow r^{3} = \frac{q Q T ^{2}}{1 6 π ^{3} ε _{0} m}