Charge \(q_2\) of mass \(m\) revolves around a stationary charge \(q_1\) in a circular orbit of radius \(r\). The orbital periodic time of \(q_2\) would be __________.

A \(\left|\frac{4 \pi^2 m r^3}{k q_1 q_2}\right|^{\frac{1}{2}}\)
B \(\left[\frac{k q_1 q_2}{4 \pi^2 m r^3}\right]^{\frac{1}{2}}\)
C \(\left[\frac{4 \pi^2 m r^4}{k q_1 q_2}\right]^{\frac{1}{2}}\)
D \(\left[\frac{4 \pi^2 m r^2}{k q_1 q_2}\right]^{\frac{1}{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\left|\frac{4 \pi^2 m r^3}{k q_1 q_2}\right|^{\frac{1}{2}}\)

Step-by-step Solution

Detailed explanation

(A)

\(\frac{m v^2}{r}=\frac{k q_1 q_2}{r^2}\)
\(\begin{aligned} & \therefore & \frac{m(r \omega)^2}{r} & =\frac{k q_1 q_2}{r^2} \\ & \therefore & \omega^2 & =\frac{k q_1 q_2}{m r^3} \\ & \therefore & \frac{4 \pi^2}{T^2} & =\frac{k q_1 q_2}{m r^3} \\ & \therefore & T^2 & =\frac{4 \pi^2 m r^3}{k q_1 q_2} \\ & \therefore & T & =\left[\frac{4 \pi^2 m r^3}{k q_1 q_2}\right]^{\frac{1}{2}}\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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