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Units and Measurements question

2024 · 1 Feb · Shift 1 · Q73
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Units and Measurements question

2024 · 1 Feb · Shift 1 · Q73

JEE MainPhysicsUnits and MeasurementsMCQ+4 / −1
The radius (r)(\mathrm{r})(r), length (l)(l)(l) and resistance (R)(\mathrm{R})(R) of a metal wire was measured in the laboratory as r=(0.35±0.05) cmR=(100±10) ohml=(15±0.2) cm\begin{aligned} & \mathrm{r}=(0.35 \pm 0.05) ~\mathrm{cm} \\\\ & \mathrm{R}=(100 \pm 10) ~\mathrm{ohm} \\\\ & l=(15 \pm 0.2)~ \mathrm{cm} \end{aligned}​r=(0.35±0.05) cmR=(100±10) ohml=(15±0.2) cm​ The percentage error in resistivity of the material of the wire is :
  1. A
    37.3%37.3 \%37.3%
  2. B
    25.6%25.6 \%25.6%
  3. C
    35.6%35.6 \%35.6%
  4. D
    39.9%39.9 \%39.9%
View written solutionFree

Correct answer: D

  1. Write the formula for resistivity

For a wire,

R=ρlAR = \rho \frac{l}{A}R=ρAl​

so

ρ=RAl=Rπr2l\rho = R\frac{A}{l} = R\frac{\pi r^2}{l}ρ=RlA​=Rlπr2​

Thus,

ρ∝R r2 l−1\rho \propto R\, r^2 \, l^{-1}ρ∝Rr2l−1
  1. Use error propagation for products and powers

For maximum percentage error,

Δρρ=ΔRR+2Δrr+Δll\frac{\Delta \rho}{\rho} = \frac{\Delta R}{R} + 2\frac{\Delta r}{r} + \frac{\Delta l}{l}ρΔρ​=RΔR​+2rΔr​+lΔl​

(Note: since lll is in the denominator, its fractional error is still added in magnitude.)

  1. Compute individual fractional errors
  • For resistance:
ΔRR=10100=0.1=10%\frac{\Delta R}{R} = \frac{10}{100} = 0.1 = 10\%RΔR​=10010​=0.1=10%
  • For radius:
Δrr=0.050.35=17≈0.142857=14.2857%\frac{\Delta r}{r} = \frac{0.05}{0.35} = \frac{1}{7} \approx 0.142857 = 14.2857\%rΔr​=0.350.05​=71​≈0.142857=14.2857%

So,

2Δrr≈28.5714%2\frac{\Delta r}{r} \approx 28.5714\%2rΔr​≈28.5714%
  • For length:
Δll=0.215=0.01333=1.333%\frac{\Delta l}{l} = \frac{0.2}{15} = 0.01333 = 1.333\%lΔl​=150.2​=0.01333=1.333%
  1. Add the percentage errors
Δρρ×100=10+28.5714+1.3333\frac{\Delta \rho}{\rho} \times 100 = 10 + 28.5714 + 1.3333ρΔρ​×100=10+28.5714+1.3333 =39.9047%≈39.9%= 39.9047\% \approx 39.9\%=39.9047%≈39.9%
  1. Match with the options
39.9%\boxed{39.9\%}39.9%​

So the correct option is D.

  1. Compare with stored answer

Stored correct answer: D

My derived answer is also D. They agree.

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