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Current Electricity question

2023 · 24 Jan · Shift 1 · Q63
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  5. /2023 · 24 Jan · Shift 1 · Q63

Current Electricity question

2023 · 24 Jan · Shift 1 · Q63

JEE MainPhysicsCurrent ElectricityNumerical+4 / −1
A hollow cylindrical conductor has length of 3.14 m, while its inner and outer diameters are 4 mm and 8 mm respectively. The resistance of the conductor is n×10−3Ωn\times10^{-3}\Omegan×10−3Ω. If the resistivity of the material is 2.4×10−8Ωm\mathrm{2.4\times10^{-8}\Omega m}2.4×10−8Ωm. The value of nnn is ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 2

  1. Use the resistance formula

For a conductor, R=ρLAR=\rho\frac{L}{A}R=ρAL​ where:

  • ρ=2.4×10−8 Ω m\rho=2.4\times 10^{-8}\,\Omega\,\text{m}ρ=2.4×10−8Ωm
  • L=3.14 mL=3.14\,\text{m}L=3.14m
  • AAA is the cross-sectional area of the hollow cylinder.
  1. Find the cross-sectional area

The conductor is hollow, so A=π(R2−r2)A=\pi\left(R^2-r^2\right)A=π(R2−r2)

Given diameters:

  • inner diameter =4 mm⇒r=2 mm=2×10−3 m=4\,\text{mm} \Rightarrow r=2\,\text{mm}=2\times 10^{-3}\,\text{m}=4mm⇒r=2mm=2×10−3m
  • outer diameter =8 mm⇒R=4 mm=4×10−3 m=8\,\text{mm} \Rightarrow R=4\,\text{mm}=4\times 10^{-3}\,\text{m}=8mm⇒R=4mm=4×10−3m

So, A=π[(4×10−3)2−(2×10−3)2]A=\pi\left[(4\times 10^{-3})^2-(2\times 10^{-3})^2\right]A=π[(4×10−3)2−(2×10−3)2] =π(16×10−6−4×10−6)=\pi\left(16\times 10^{-6}-4\times 10^{-6}\right)=π(16×10−6−4×10−6) =π⋅12×10−6=\pi\cdot 12\times 10^{-6}=π⋅12×10−6 =12π×10−6 m2=12\pi\times 10^{-6}\,\text{m}^2=12π×10−6m2

  1. Substitute into resistance formula

R=2.4×10−8×3.1412π×10−6R=\frac{2.4\times 10^{-8}\times 3.14}{12\pi\times 10^{-6}}R=12π×10−62.4×10−8×3.14​

Using 3.14≈π3.14\approx \pi3.14≈π, R=2.4×10−8×π12π×10−6R=\frac{2.4\times 10^{-8}\times \pi}{12\pi\times 10^{-6}}R=12π×10−62.4×10−8×π​ =2.412×10−2=\frac{2.4}{12}\times 10^{-2}=122.4​×10−2 =0.2×10−2=0.2\times 10^{-2}=0.2×10−2 =2×10−3 Ω=2\times 10^{-3}\,\Omega=2×10−3Ω

  1. Compare with the given form

Given, R=n×10−3 ΩR=n\times 10^{-3}\,\OmegaR=n×10−3Ω So, n=2n=2n=2

Final Answer: 2\boxed{2}2​

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