When two spheres having \(4 Q\) and \(-2 Q\) charge are placed at a certain distance, the force acting between them is F. Now they are connected by a conducting wire and again separated from each other. Now they are kept at a distance half of the previous one. The force acting between them is ________.

A \(\frac{ F }{2}\)
B F
C \(\frac{ F }{4}\)
D \(\frac{ F }{8}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{ F }{2}\)

Step-by-step Solution

Detailed explanation

(A)
\(
\begin{array}{l}
F=\frac{k(4 Q)(2 Q)}{r^2} \\
F=\frac{8 k Q^2}{r^2} \text { (initial force) } \quad \quad \ldots \ldots(1)
\end{array}
\)
When they are connected by a conducting wire and again separated from each other, then charge on each sphere.
\(\begin{array}{l}=\frac{4 Q-2 Q}{2} \\ =\frac{2 Q}{2} \\ =Q\end{array}\)
When they are kept at a distance half of the previous one, coloumbian force between them is
\(\begin{array}{l} F ^{\prime}=\frac{k( Q )( Q )}{\left(\frac{r}{2}\right)^2} \\ F^{\prime}=\frac{4 k Q ^2}{r^2}=\frac{8 k Q ^2}{2 r^2} \\ F^{\prime}=\frac{ F }{2}\end{array}\)

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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