Question

The resistivity of a material depends on relaxation time tau of electrons,charge on electrons,no density of electron and mass of electron. find the expression for resistivity of the material.

Answer 1

$$\begin{gathered} \mathrm{Drift}\mathrm{velocity}\left(v_{\mathrm{d}}\right)\mathrm{is}\mathrm{given}\mathrm{by} \\ {v}_d=\frac{eE}{m}\tau \\ \Rightarrow v_d=\frac{eV}{ml}\tau \left(\because E=\frac{V}{l}\right) \\ \mathrm{Relation}\mathrm{between}\mathrm{drift}\mathrm{velocity}\mathrm{and}\mathrm{current}\mathrm{is} \\ I=Ane{v}_{\mathrm{d}} \\ \therefore \mathit{}I=Ane\left(\frac{eV}{ml}\tau \right)=\left(\frac{An{e}^2\tau }{ml}\right)V \\ \mathit{\Rightarrow }\frac{V}{I}=\frac{ml}{An{e}^2\tau }=R.....\left(1\right) \\ \mathrm{Resistance}\mathrm{is}\mathrm{also}\mathrm{given}\mathrm{by} \\ R\mathit{=}\rho \frac{\mathit{l}}{\mathit{A}}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\mathit{.}\left(\mathit{2}\right) \\ \mathrm{Comparing}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get} \\ \rho \mathit{=}\frac{\mathit{m}}{\mathit{n}{\mathit{e}}^{\mathit{2}}\mathit{\tau }} \end{gathered}$$