Are spherical coordinates distances or angles?

Question

I've become confused about spherical coordinates when dealing with electric fields.

The way I always understood spherical coordinates is something like the below picture. To define a vector, you give it a distance outwards (r), and two angles to get a final position. Below, the $\theta$ and $\phi$ components are measured in radians.

(Courtesy Wikipedia.org)

However, you can also have, say, an electric field in spherical coordinates. In this case, the unit vectors $\theta$ and $\phi$ don't define angles but rather values of the vector fields. So, in the case of electric fields, we might have $E_\theta = 10\text{ Vm}$. That is, at every point there will be this electric field component in the theta direction.

So, it seems there are two different ways of dealing with spherical coordinates. One, where the $\theta$ and $\phi$ components represent angles, and one where they represent values of the components in those directions.

This would then give you two different measures of lengths of the vectors. In the first case, the length of the vector is always given by the r component. In the second case, you take $|\vec{E}|=\sqrt{E_r^2+E_\theta^2+E_\phi^2}$.

What am I mixing up here?

Answer 1

You can't take the root mean square of the spherical coordinate parameters because they aren't all the same units (one is a length measurement while the other two are angle values). Well you can, but the output is meaningless.

To convert spherical parameters to Cartesian coordinates, you use simple trig:

\begin{align}E_z & = r \cos(\theta) \\ E_x & = r \sin(\theta)\cos(\phi) \\ E_y &= r \sin(\theta)\sin(\phi) \end{align}

Now if you take the root mean square of $E_x$, $E_y$ and $E_z$ you will get the correct $r$.

Answer 2

For an electric field usually you have three components like $(E_x, E_y, E_z)$ in Cartesian coordinate system. Now you want to rewrite the same vector in a spherical coordination, what you should do is as follows: first you write the vector like the electric field as $\mathbf{E}=|E|\mathbf{e_r}$ where $|E|$ is given by the $|E|=\sqrt{E_x^2+E_y^2+E_z^2}$ and $\mathbf{e_r}$ is a function of $\theta$ and $\phi$. Then One should have $\theta=\text{acos}(E_z/|E|)$ and $\phi=\text{atan}(E_y/E_x)$. If you like to write in a three component form, it is always (|E|, 0, 0).