Question 2 Solution
1. Resistance Formula:
The resistance of a wire is given by R = ρ L / A.
The resistance of a wire is given by R = ρ L / A.
2. Logarithmic Differentiation:
To find the rate of change with temperature, we take the natural log of both sides and then differentiate:
ln(R) = ln(ρ) + ln(L) – ln(A)
(1/R)(dR/dT) = (1/ρ)(dρ/dT) + (1/L)(dL/dT) – (1/A)(dA/dT)
To find the rate of change with temperature, we take the natural log of both sides and then differentiate:
ln(R) = ln(ρ) + ln(L) – ln(A)
(1/R)(dR/dT) = (1/ρ)(dρ/dT) + (1/L)(dL/dT) – (1/A)(dA/dT)
3. Coefficients of Expansion:
Let αρ be the temp. coefficient of resistivity and αL be the linear expansion coefficient.
(1/R)(dR/dT) = αρ + αL – 2αL
(1/R)(dR/dT) = αρ – αL
Let αρ be the temp. coefficient of resistivity and αL be the linear expansion coefficient.
- (1/ρ)(dρ/dT) = αρ
- (1/L)(dL/dT) = αL
- Since Area A ∝ L2, its coefficient is 2αL.
(1/R)(dR/dT) = αρ + αL – 2αL
(1/R)(dR/dT) = αρ – αL
4. Condition for Least Effect:
For resistance to be independent of temperature, dR/dT should be zero.
0 = αρ – αL &implies; αρ = αL
For resistance to be independent of temperature, dR/dT should be zero.
0 = αρ – αL &implies; αρ = αL
Correct Option: (c) Temperature coefficient of resistivity is equal to that of linear expansion.