निम्नलिखित में से कौन से समीकरण विमीय रूप से सत्य हैं ? जहाँ \(t =\) समय, \(h =\) ऊँचाई, \(s =\) पष्ठ तनाव, \(\theta=\) कोण, \(\rho=\) घनत्व, \(a , r =\) त्रिज्या, \(g =\) गुरूत्वीय त्वरण, \(v =\) आयतन, \(p =\) दाब, \(W =\) किया गया कार्य, \(\Gamma=\) बल आधूर्ण, \(\varepsilon=\) विद्युत शीलता, \(E =\) विद्युत क्षेत्र, \(J =\) धारा घनत्व, \(L =\) लंबाई।

A \({v}=\frac{\pi {pa}^{4}}{8 \eta {L}}\)
B \({h}=\frac{2 {s} \cos \theta}{\rho {rg}}\)
C \({J}=\varepsilon \frac{\partial {E}}{\partial {t}}\)
D \({W}=\Gamma \theta\)

Verified Solution

Answer & Solution

Correct Answer

(A) \({v}=\frac{\pi {pa}^{4}}{8 \eta {L}}\)

Step-by-step Solution

Detailed explanation

\((i)\) \(\frac{\pi {pa}^{4}}{8 \eta {L}}=\frac{{d} {v}}{{dt}}=\) Volumetric flow rate (poiseuille's law) \((ii)\) \({h} \rho {g}=\frac{2 {s}}{{r}} \cos \theta\) \((iii)\)…

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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