Varying Volume Charge Density in a Solid Cylinder

  • #1

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 [itex]\rho(z)=\rho_0+\beta z[/itex]. Find the electric field at the origin (in terms of R, L, [itex]\rho_0[/itex], and [itex]\beta[/itex].

Homework Equations



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

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


The Attempt at a Solution



In cylindrical coordinates, [itex]\vec{\eta}=\vec{r'}=s'\hat{s'}+\phi'\hat{\phi'}+z'\hat{z'}[/itex]. 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: [itex] 0\leq s'\leq R, 0\leq \phi'\leq 2\pi, -L/2\leq z\leq L/2[/itex]? On a side note, is it better to use Gauss' Law here?

Thanks.
 

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