Varying Volume Charge Density in a Solid Cylinder

[BlackHole213]

Homework Statement

Consider a right circular cylinder with radius R and height L oriented along the z-axis. The center of the cylinder coincides with the origin. Inside the cylinder the volume charge density is given by $\rho(z)=\rho_0+\beta z$. Find the electric field at the origin (in terms of R, L, $\rho_0$, and $\beta$.

Homework Equations

$\vec{E(\vec{r})}=\frac{1}{4\pi\epsilon_0}\int_a^b \! \rho(\vec{r'})\frac{\hat{\eta}}{\eta^2} \mathrm{d}\tau'.$

where $\vec{\eta}=\vec{r}-\vec{r'}$ and r is the field point and r' is the source point. Thus, in this problem, $\vec{r}$=0, $\eta=-\vec{r'}$ and $\eta^2=|\vec{r'}|^2$. Everything with a prime on it corresponds to r'.

The Attempt at a Solution

In cylindrical coordinates, $\vec{\eta}=\vec{r'}=s'\hat{s'}+\phi'\hat{\phi'}+z'\hat{z'}$. As we want to know the electric field at the origin, in the volume charge density equation, would I set z=0, or would I set z=z' and then use the electric field integral formula integrating: $0\leq s'\leq R, 0\leq \phi'\leq 2\pi, -L/2\leq z\leq L/2$? On a side note, is it better to use Gauss' Law here?

Thanks.

 

[mark726]

To find the electric field at the origin, you would set z=0 in the volume charge density equation. This is because the electric field at the origin is only affected by the charge density at that point, not at any other point within the cylinder. Therefore, the integral would become:

\vec{E(0)}=\frac{1}{4\pi\epsilon_0}\int_a^b \! (\rho_0+\beta z')\frac{\hat{\eta}}{\eta^2} \mathrm{d}\tau'.

Since we are only considering the charge density at the origin, we can simplify the integral to:

\vec{E(0)}=\frac{1}{4\pi\epsilon_0}\int_a^b \! \rho_0\frac{\hat{\eta}}{\eta^2} \mathrm{d}\tau'.

As for using Gauss' Law, it is certainly a valid approach to finding the electric field in this scenario. However, using the integral formula provided in the problem statement would also work. It ultimately depends on your comfort level with each method and which one you feel most confident using. Both should yield the same result. I hope this helps.