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Physics

Given that the resistivity of Gold and Platinum is 2.1 x 10-8 ohm.m and 10.5 x 10-8 ohm.m, respectively, which is the correct statement?

Statement A : 5 metres of platinum wire will have the same resistance as one metre of gold wire.

Statement B : Gold wire of 1 cm radius will have the same resistance as platinum wire of 5 cm radius.

  1. Only statement A is correct.
  2. Only statement B is correct.
  3. Both statements A and B are correct.
  4. Both statements A and B are incorrect.

Current Electricity

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Answer

Both statements A and B are incorrect.

Explanation :

By resistance formula,
R=ρ⋅LA\text{R} = \rho \cdot \dfrac{\text{L}}{\text{A}}

Statement (A) is incorrect:
Given,
Resistivity of gold ρAu= 2.1 x 10-8 ohm.m
Resistivity of platinum ρPt= 10.5 x 10-8 ohm.m

Let the cross-sectional area A be the same for both wires.

Resistance of gold:

RAu=ρAulAuARAu=2.1×10−8×1ARAu=2.1×10−8AR{Au}=ρ{Au} \dfrac{l{Au}}{A} \\[1em] R{Au}=2.1\times10^{-8} \times\dfrac{1}{A} \\[1em] R_{Au}= \dfrac{2.1\times10^{-8}}{A} \\[1em]

Resistance of platinum:

RPt=ρPtlPtARPt=10.5×10−8×5ARPt=52.5×10−8AR{Pt}=ρ{Pt} \dfrac{l{Pt}}{A} \\[1em] R{Pt}=10.5\times10^{-8} \times\dfrac{5}{A} \\[1em] R_{Pt}= \dfrac{52.5\times10^{-8}}{A} \\[1em]

RAuR{Au} ≠ RPtR{Pt}

Statement (B) is incorrect:
Let the length of both wires are same.
Resistance of gold:

RAu=ρAulAAuRAu=2.1×10−8×lπ×12RAu=2.1×10−8×lπR{Au}=ρ{Au} \dfrac{l}{A{Au}} \\[1em] R{Au}=2.1\times10^{-8} \times\dfrac{l}{π\times 1^2 } \\[1em] R_{Au}= \dfrac{2.1\times10^{-8}\times l}{π} \\[1em]

Resistance of platinum:

RPt=ρPtlAPtRPt=10.5×10−8×lπ×52RAu=10.5×10−8×l25πR{Pt}=ρ{Pt} \dfrac{l}{A{Pt}} \\[1em] R{Pt}=10.5\times10^{-8} \times\dfrac{l}{π\times 5^2 } \\[1em] R_{Au}= \dfrac{10.5\times10^{-8}\times l}{25π}

RAuR{Au} ≠ RPtR{Pt}

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