UY1: Resistance Of A Cylindrical Resistor



A hollow cylinder has length L and inner and outer radii a and b. It is made of a material with resistivity $\rho$. A potential difference is set up between the inner and outer surfaces of the cylinder, each of which is an equipotential surface) so that current flows radially through the cylinder. What is the resistance to this radial current flow?

We recall that $J = \frac{I}{A}$, $E = \frac{dV}{dr}$. The area, A will be the surface area of the cylinder. Since

$$\begin{aligned} J &= \frac{1}{\rho} E \\ \frac{I}{A} &= \frac{1}{\rho} E \\ \frac{I}{2 \pi r L} &= \frac{1}{\rho} \frac{dV}{dr} \\ dV &= I \frac{\rho}{2 \pi r L} \, dr \end{aligned}$$

Since $dR = \frac{dV}{I}$,

$$ dR = \frac{\rho}{2 \pi r L} \, dr $$

Integrating from a to b,

$$\begin{aligned} R &= \int\limits_{a}^{b} dR \\ &= \int\limits_{a}^{b} \rho \frac{1}{2 \pi r L} \, dr \\ &= \frac{\rho}{2 \pi L} \int\limits_{a}^{b} \frac{1}{r} \, dr \\ &= \frac{\rho}{2 \pi L} \, \text{ln} \, \frac{b}{a} \end{aligned}$$

 

 

Next: Electromotive Force & Power In Circuits

Previous: Resistance And Resistivity

Back To Electromagnetism (UY1)

 

Back To University Year 1 Physics Notes



Mini Physics

As the Administrator of Mini Physics, I possess a BSc. (Hons) in Physics. I am committed to ensuring the accuracy and quality of the content on this site. If you encounter any inaccuracies or have suggestions for enhancements, I encourage you to contact us. Your support and feedback are invaluable to us. If you appreciate the resources available on this site, kindly consider recommending Mini Physics to your friends. Together, we can foster a community passionate about Physics and continuous learning.



Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.