Dipole moment of a cylinder

Tags:
1. Nov 10, 2014

precise

1. The problem statement, all variables and given/known data
I am trying to calculate the dipole moment of a cylinder of volume charge density $\rho_0$ of radius $R$ and height $H$ with is center coinciding with the origin. My guess is that it should be 0 because of the symmetry but I am not able to show it. Below is my calculation attempt. Thanks for any help.

2. Relevant equations
$$P=\int_V \vec r \rho(\vec r)d\tau$$
3. The attempt at a solution
$$P=\int_V \vec r \rho(\vec r)d\tau = \rho_0 \int_0^R \int_0^{2\pi} \int_{-H/2}^{H/2} (r e_{\hat r} + ze_{\hat z}) rdrd\theta dz = 2\pi \rho_0 (\frac {R^3 H} {3}e_\hat r + 0e_{\hat z})$$

2. Nov 15, 2014

Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 15, 2014

TSny

Hello.
Looks like you are treating the unit vector $\hat{e}_r$ as a constant vector in the integration. Does it have the same direction throughout the cylinder?

What is the meaning of $\hat{e}_r$ in your final answer? What direction does this $\hat{e}_r$ point?

Can you express $\hat{e}_r$ at a point in the cylinder in terms of $\hat{e}_x$, $\hat{e}_y$, and $\theta$? Would this help?