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  • A cylindrical conductor of length  and inner radius R1 and outer radius R2 has specific resistance . A

A cylindrical conductor of length \ell and inner radius R1 and outer radius R2 has specific resistance \rho. A cell of emf \varepsilon is connected across the two lateral faces of the conductor. Find the current drown from the cell.  
        

Option: 1

I=\frac{2 \pi l \varepsilon}{\rho \ln \left(\frac{R_2}{R_1}\right)}


Option: 2

I=\frac{2 \pi \varepsilon}{\rho \ln \left(\frac{R_1}{R_2}\right)}


Option: 3

\mathrm{I}=\frac{2 \pi l \varepsilon}{\rho \ln \left(\frac{R_2}{R_1}\right)^2}


Option: 4

\mathrm{I}=\frac{2 \pi}{\rho \ln \left(\frac{R_2}{R_1}\right)}


Answers (1)

best_answer

Consider the differential element of the cylinder as shown in the figure.
\therefore \quad \mathrm{R}=\int_{\mathrm{R}_1}^{\mathrm{R}_2} \rho \frac{\mathrm{dx}}{2 \pi \mathrm{xl}} \quad\left(\because \mathrm{R}=\rho \frac{l}{\mathrm{a}}\right)
\begin{aligned} \Rightarrow \quad \mathrm{R} & =\frac{\rho}{2 \pi l} \ln \left(\mathrm{R}_2 / \mathrm{R}_1\right) \\ \mathrm{I} & =\frac{\varepsilon}{\mathrm{R}} \\ \Rightarrow \quad \mathrm{I} & =\frac{2 \pi l \varepsilon}{\rho \ln \left(\frac{\mathrm{R}_2}{\mathrm{R}_1}\right)} \end{aligned}

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