MHT CET · Physics · Electrostatics

Four point charges each \(+q\) is placed on the circumference of a circle of diameter 2 d in such a way that they form a square. The potential at the centre is proportional to

A \(\frac{\mathrm{q}^2}{\mathrm{~d}^2}\)
B \(\frac{q}{d}\)
C \(\frac{d}{q}\)
D \(\frac{\mathrm{d}^2}{\mathrm{q}^2}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{q}{d}\)

Step-by-step Solution

Detailed explanation

As charges are placed on the circumference of circle, each charge is at a distance,
\(\mathrm{r}=\frac{\text { Diameter }}{2}=\frac{2 \mathrm{~d}}{2}=\mathrm{d}\)
\(\therefore \quad\) Potential at centre due to four point charges
\(\begin{aligned}
& =\frac{1}{4 \pi \varepsilon_0}\left[\frac{\mathrm{q}}{\mathrm{~d}}+\frac{\mathrm{q}}{\mathrm{~d}}+\frac{\mathrm{q}}{\mathrm{~d}}+\frac{\mathrm{q}}{\mathrm{~d}}\right] \\
& =\frac{1}{4 \pi \varepsilon_0} \frac{4 \mathrm{q}}{\mathrm{~d}}=4 \dot{K}\left[\frac{\mathrm{q}}{\mathrm{~d}}\right]
\end{aligned}\)
Potential \(\propto \frac{q}{d}\)

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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