a
we will find the relation for small temperature changes.

resistance \(R =\rho L / A\)

coefficient of linear expansion \(= \alpha\)

length of conductor: \(L = L _0(1+ a \Delta T ) \quad \Delta L = a L _0 \Delta T\)

\(\beta=\) coefficient of expansion in area : \(=2 \alpha\)

Area of cross section: \(A=A_0(1+2 \alpha \Delta T)\)

\(\Delta A=2 \alpha A_0 \Delta T\)

Resistivity \(\rho=\rho_0(1+\alpha_p \Delta T)\)

\(\Delta \rho=\rho_0 \alpha_p \Delta T\)

Resistance \(R = R _0(1+\alpha_R \Delta T )\)

\(\Delta R =\alpha_R R _0 \Delta T\)

If \(\Delta A =2 a \Delta T\) is very small then, and for small \(\Delta T\),

\(R_0=\rho_0 L_0 / A_0\)

\(R =\rho_0(1+ \alpha_p \Delta T ) L _0(1+ a \Delta T ) /\left[ A _0(1+2 \alpha \Delta T )\right]\)

\(=\left(\rho_0 L_0 / A_0\right)(1+\alpha_p \Delta T)(1+\alpha \Delta T)(1-2 \alpha \Delta T)\)

\(=R_0(1+\alpha_p \Delta T)(1-a \Delta T) \quad\) ignoring the \(2 a^2 \Delta T^2\) term

\(=R_0[1+(\alpha_p-\alpha) \Delta T] \quad\) ignoring the \(A \rho a \Delta T^2\) term

\(\alpha_R=\alpha_P-\alpha\)