किसी द्रव्यमान वितरण के कारण \(x\)-अक्ष पर मूलबिन्दु से \(x\) दूरी पर \(x\)-दिशा में गुरूत्वाकर्षण क्षेत्र \(\frac{ Ax }{\left( x ^{2}+ a ^{2}\right)^{3 / 2}}\) द्वारा दिया जाता है। \(x\)-अक्ष पर \(x\) दूरी पर गुरूत्वीय विभव का परिमाण क्या होगा जबकि अनन्त पर इसका मान शून्य लिया गया है।

A \(\frac{A}{\left(x^{2}+a^{2}\right)^{1 / 2}}\)
B \(\frac{A}{\left(x^{2}+a^{2}\right)^{3 / 2}}\)
C \(A \left( x ^{2}+ a ^{2}\right)^{3 / 2}\)
D \(A\left(x^{2}+a^{2}\right)^{1 / 2}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{A}{\left(x^{2}+a^{2}\right)^{1 / 2}}\)

Step-by-step Solution

Detailed explanation

Given \(E _{ G }=\frac{ Ax }{\left( x ^{2}+ a ^{2}\right)^{3 / 2}}, V _{\infty}=0\) \(\int_{V_{\infty}}^{V_{x}} d V=-\int_{\infty}^{x} \vec{E}_{G} \cdot \vec{d}_{x}\) \(V _{ x }- V _{\infty}=-\int_{\infty}^{ x } \frac{ Ax }{\left( x ^{2}+ a ^{2}\right)^{3 / 2}} d x\) put…

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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