Finding the Dipole Moment of a Cylinder

[precise]

Homework Statement

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.

Homework Equations

$$ P=\int_V \vec r \rho(\vec r)d\tau
$$

The Attempt at a Solution

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

 

[TSny]

Hello.

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

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?