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Cylindrical coordinates of a ring

Physics Asked on July 22, 2021

A classical problem for electromagnetism students is the calculation of the electric field on the central axis of a ring. It can be solved in many different ways, but I got stuck with the pure integration method. Let me introduce you the steps written in my book.

The electric field in an arbitrary point $mathbf{r}$ of the space, is given by the following expression:
$$
mathbf{E}(mathbf{r}) = dfrac{1}{4piepsilon_0} int_{L}{dfrac{lambda(mathbf{r’})(mathbf{r}-mathbf{r’})dl}{left| mathbf{r}-mathbf{r’} right|^3}}
$$

Then, in order to solve the integral, it says that in the central axis, the point in which we are calculating the field is given by $mathbf{r} = z,mathbf{u_z}$, which seems ok. However, the part where I get lost is regarding the position of each elemental charge $lambda(mathbf{r’})dl$, where the book says that this is $mathbf{r’} = R,mathbf{u_rho}$, where $R$ is the radius of the ring.

This also looks ok to me, except that not every differential charge is on the same $varphi$ cylindrical coordinate. So I don’t get why $mathbf{r’}$ has not the form $R,mathbf{u_rho} + varphi,mathbf{u_varphi}$.

And then, given the previous question, what would be the proper way of expressing a vector $(rho,varphi,z)$ as a sum of the basis and why?
$$
mathbf{v} = rho,mathbf{u_rho} + varphi,mathbf{u_varphi} + z,mathbf{u_z} quad text{or} quad mathbf{v} = rho,mathbf{u_rho} + rhovarphi,mathbf{u_varphi} + z,mathbf{u_z}
$$

2 Answers

I met this issue before, I guess you are using wikipedia's notion. Notice that $(r-r')=|r-r'|hat{r}$ and you can reduce the complicated notion to $1/r^2 hat{r}$. The last question, just google cylinder coordinates, either wolfram or Wikipedia has tons' of explanations. (Wolfram might be very technical, so you might consider Wikipedia, https://en.wikipedia.org/wiki/Cylindrical_coordinate_system ) Just to be clear,$hat{e}_varphi,hat{e}_rho,hat{e}_z$ were the orthogonal basis, which express the 3 space completely if you accept $x,y,z$ as a complete expansion. The proper proof comes from Lagrangian's generalized motion.

Answered by J C on July 22, 2021

You will get in trouble if you try to apply angular basis vectors for radial vectors coming out of the origin.

It is best to form cartesian vectors.

$overset{rightharpoonup }{r}'=left{R cos left(phi 'right),R sin left(phi 'right),0right}$

$overset{rightharpoonup }{r}={0,0,z}$

for a ring of charge centered in the $xy$ plane and an observation point on the z axis.

You can still integrate over $d overset{rightharpoonup }{l}=R text{d$phi $} $

Answered by Bill Watts on July 22, 2021

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