Question

Find the electric field due to a point charge in cylindrical-polar coordinates.

Answer 1

\[\vec{E} = \frac{q}{4 \pi \epsilon_0} \frac{\vec{r}}{|\vec{r}|^3}\]
I figure \(|\vec{r}|=\sqrt{x^2+y^2+z^2}\) and \(x^2+y^2=p^2\)
so \(|\vec{r}|^3=(p^2+z^2)^{\frac{3}{2}}\)

But I have no idea what to do with the \(\vec{r}\) on the top

Answer 2

Do I rewrite it in term of the cylindrical coordinate unit vectors? \(\vec{r}=p\hat p + \phi\hat \phi + z\hat z\)
But this really doesn't make sense to me, do I even need to include the \(\phi\)? or what simplifications can be done? it's symmetric over all angles of \(\phi\) right? so something should be able to get simplified but i'm not quite sure.

Answer 3

I think that we have to start from the potetial function, namely:
\[\varphi \left( r \right) = \frac{q}{r}\]
where q is the charge

Answer 4

then you have to apply the subsequent equation:
\[{\mathbf{E}} = - \nabla \varphi \]
using the cylindrical coordinate

Answer 5

in cylindrical coordinate, we have the subsequent identity:
\[\nabla \psi = \left( {\frac{{\partial \psi }}{{\partial r}},\;\frac{1}{r}\frac{{\partial \psi }}{{\partial \theta }},\;\frac{{\partial \psi }}{{\partial z}}} \right)\]
where: \[r,\;\theta {\text{ and }}z\]
are the cylindrical coordinates