एक बिन्दु द्वि- ध्रुव \(\overrightarrow{ p }=- p _{0} \hat{ x }\) मूल बिन्दु पर स्थित है। तो इस द्विध्रुव के कारण, \(y\)-अक्ष पर \(d\) दूरी पर, विभव तथा विधुत क्षेत्र होंगे क्रमशः (मानो अनंत पर \(V =0\) है):

A \(\frac{{\left| {\vec p} \right|}}{{4\pi {\varepsilon _0}{d^2}}},\frac{{ -\vec p}}{{4\pi {\varepsilon _0}{d^3}}}\)
B \(0,\frac{{ \vec p}}{{4\pi {\varepsilon _0}{d^3}}}\)
C \(\frac{{\left| {\vec p} \right|}}{{4\pi {\varepsilon _0}{d^2}}},\frac{{ \vec p}}{{4\pi {\varepsilon _0}{d^3}}}\)
D \(0,\frac{{ - \vec p}}{{4\pi {\varepsilon _0}{d^3}}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(0,\frac{{ - \vec p}}{{4\pi {\varepsilon _0}{d^3}}}\)

Step-by-step Solution

Detailed explanation

\(V=0\) \(E=-\frac{K \vec{P}}{r^{3}}\) \(=-\frac{\vec{p}}{4 \pi \varepsilon_{0} d^{3}}\)

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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