Current Electricity

JEE Main level, single correct, medium

89 of 380

Question 89 of 380

A material ' BB ' has twice the specific resistance of ' AA '. A circular wire made of ' BB ' has twice the diameter of a wire made of ' AA '. Then for the two wires to have the same resistance, the ratio ℓA/ℓB\ell_A / \ell_B of their respective lengths must be

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Answer: (A) 2

R1=ρAℓAπRA2R2=ρBℓBπRB2\mathrm{R}_1=\frac{\rho_{\mathrm{A}} \ell_{\mathrm{A}}}{\pi \mathrm{R}_{\mathrm{A}}^2} \quad \mathrm{R}_2=\frac{\rho_{\mathrm{B}} \ell_{\mathrm{B}}}{\pi \mathrm{R}_{\mathrm{B}}^2}
ℓAℓB=ρBRA2ρARB2=2ρARA2ρA⋅4RA2⇒ℓBℓA=2\frac{\ell_{\mathrm{A}}}{\ell_{\mathrm{B}}}=\frac{\rho_{\mathrm{B}} \mathrm{R}_{\mathrm{A}}^2}{\rho_{\mathrm{A}} \mathrm{R}_{\mathrm{B}}^2}=\frac{2 \rho_{\mathrm{A}} \mathrm{R}_{\mathrm{A}}^2}{\rho_{\mathrm{A}} \cdot 4 \mathrm{R}_{\mathrm{A}}^2} \Rightarrow \frac{\ell_{\mathrm{B}}}{\ell_{\mathrm{A}}}=2

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