Electric field due to point charge in cylindrical co-ordinates

[fengqiu]

I need to use coulombs law to describe the electric field due to a point charge in cylindrical co ordinates. I know the answer should have E = [ρ,0,z] with the azimuth as 0 but I can't show it using the standard electric field equations. please note I need to use E=q/4πε * r/|r|^3 I'm sorry I don't know how to type it all pretty in latex :( it's on my to do list
E=q/4πε * r/|r|^3
I'm truly perplexed on where to start
Thanks for all the help!

 

[BruceW]

hey, welcome to physicsforums :)
well, start writing down the equations you know will be useful. You know the answer in Cartesian coordinates. So then the question is how to get to cylindrical coordinates, from the Cartesian ones. Also, it seems they want you to give an answer in terms of cylindrical unit vectors, so you're going to need to decide what the cylindrical unit vectors are.

 

[fengqiu]

Hi thanks for the help. So say in the simple case i have E=q/4pie0 * r/(|r|^3), so basically i use the unit basis vectors as r/|r| = rhat. now from common sense I know that r = p^2+z^2 but to prove this mathematically...
I think phi(hat) = -xhat * sin(phi) +yhat*cos(phi) which.. makes me stuck haha,
how does this equal 0?

thanks loads BruceW for your help! I feel so nooby, i guess that's what i get from jumping straight back intp physics after an engineering background!

 

[vanhees71]

The field is
$$\vec{E}=\frac{Q}{4 \pi \epsilon_0 r^3} \vec{r},$$
and you can easily rewrite everything in cylinder coordinates, e.g.,
$$\vec{r}=\rho \vec{e}_{\rho} + z \vec{e}_z$$
etc.

 

[paranormal]

No problem, I can assist you with this. The electric field due to a point charge in cylindrical coordinates can be calculated using Coulomb's Law, just as you mentioned. First, let's define our coordinate system. In cylindrical coordinates, we have three variables: r, θ, and z.

Now, we know that the electric field at any point is given by the formula E=q/4πε * r/|r|^3. In this equation, q represents the charge of the point charge, ε represents the permittivity of the medium, and r represents the distance from the point charge to the point where we want to calculate the electric field.

In cylindrical coordinates, we can express the distance r as r=√(x^2 + y^2). Since we are dealing with a point charge, we can assume that the charge is located at the origin, so x=0 and y=0. This means that r=√(0^2 + 0^2)=0. Therefore, the electric field equation becomes E=q/4πε * 0/|0|^3=0.

Now, we need to consider the direction of the electric field. In cylindrical coordinates, the electric field vector is given by E=[E_r, E_θ, E_z]. We know that the electric field is radial, meaning it points outward from the origin. This means that the electric field vector has no θ component (E_θ=0) and only has a r component (E_r=q/4πε * r/|r|^3). Therefore, the electric field in cylindrical coordinates is E=[q/4πε * r/|r|^3, 0, 0].

I hope this helps to clarify the electric field due to a point charge in cylindrical coordinates. Let me know if you have any further questions.