JEE Mains · Physics · STD 12 - Current electricity
- JEE Mains 2019
- Hard
- MCQ
In a meter bridge, the wire of length \(1\, m\) has a nonuniform cross-section such that, the variation \(\frac{{dR}}{{d\ell }}\) of its resistance \(R\) with length \(\ell \) is \(\frac{{dR}}{{d\ell }} \propto \frac{1}{{\sqrt \ell }}\) Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point \(P\). What is the length \(AP\) ? ................ \(m\)
Pick an option to check your answer.
Answer & Solution
Correct Answer
(C) \(0.25\)
Step-by-step Solution
Detailed explanation
\(\frac{\mathrm{d} \mathrm{R}}{\mathrm{d} \ell}=\frac{\mathrm{k}}{\sqrt{\ell}} \quad \mathrm{k}=\mathrm{constant}\)
\(\int_{0}^{R} d R=k \int_{0}^{1} \frac{d \ell}{\sqrt{\ell}}\)
\(\mathrm{R}=2 \mathrm{k}\) resistance of wire \(\mathrm{AB}\)
Again, \(\int_{0}^{R / 2} \mathrm{d} \mathrm{R}=\mathrm{k} \int_{0}^{L} \frac{\mathrm{d} \ell}{\sqrt{\ell}} \quad \mathrm{L} \rightarrow\) Length \(\mathrm{AP}\)
\(\frac{\mathrm{R}}{2}=\mathrm{k} 2 \mathrm{L}^{1 / 2} \quad ; \quad \mathrm{k}=\mathrm{k} 2 \mathrm{L}^{1 / 2}\)
\(\Rightarrow \quad \mathrm{L}=\frac{1}{4}\, \mathrm{m}=0.25 \,\mathrm{m}\)
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