कुल आवेश \(2 Q\) को त्रिज्या \(R\) के गोले में इस प्रकार वितरित करते हैं कि आवेश घनत्व सम्बन्ध \(\rho( r )= kr\) से दिया जाता है जहाँ \(r\), केन्द्र से दूरी है। दो बराबर \(Q\) आवेशों \(A\) तथा \(B\) को केन्द्र से \(a\) दूरी पर व्यासीय विपरीत बिन्दुओं पर रखा गया है। यदि \(A\) और \(B\) कोई बल अनुभव नहीं करते हैं, तो ?

A \(a = \frac{{3R}}{{{2^{1/4}}}}\)
B \(a = {2^{ - 1/4}}R\)
C \(a = {8^{ - 1/4}}R\)
D \(a = R/\sqrt 3 \)

Verified Solution

Answer & Solution

Correct Answer

(C) \(a = {8^{ - 1/4}}R\)

Step-by-step Solution

Detailed explanation

\(E 4 \pi a^{2}=\frac{\int_{0}^{\theta} k r 4 \pi r^{2} d r}{e_{0}}\) \(E=\frac{k 4 \pi a^{4}}{4 \times 4 \pi \varepsilon_{0}}\) \(2 \mathrm{Q}=\int_{0}^{\mathrm{R}} \mathrm{kr} 4 \pi \mathrm{r}^{2} \mathrm{dr}\) \(\mathrm{k}=\frac{2 \mathrm{Q}}{\pi \mathrm{R}^{4}}\)…

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.

Same subject

Explore more questions on app

From JEE Mains

Explore more questions on app