Express E due to a point charge at the origin

In summary, The electric field E due to a point charge q at the origin can be expressed in cylindrical polar coordinates as $$\vec E = {q\over 4\pi\varepsilon_0 r^2} \, \hat r + 0 \, \hat \phi + 0 \, \hat \theta$$. This is obtained by converting the expression in spherical coordinates using the linear transformations for converting from spherical to cylindrical coordinates. It is important to note that the radial component of the electric field, E_r, is the only component that depends on the distance from the origin, r. The other components, E_omega and E_theta, are both equal to zero.
  • #1
leeban7
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Homework Statement


Express the electric field E due to a point charge q at the origin in cylindrical polar coordinates.

Homework Equations

The Attempt at a Solution


Know that E = q / 4*pi*epsilon_0*r^2 in the r-direction, which is the answer in spherical coordinates. How we we swap to cylindrical?
 
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  • #2
Hello leeban, :welcome:

Same as when we would have to change to Cartesian coordinates:
We simply express the ##\vec E## in terms of ##E_\rho##, ##E_\phi## and ##E_z## :smile:
 
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  • #3
Do you recommend swapping to Cartesian and then to cylindrical? I know the linear transforms for those, but not from spherical to cylindrical
 
  • #4
I do not recommend that. For one, you already have that ##\phi## in the one is equal to ##\phi## in the other system (*)
And for the ##z##, yes, you have to do that, because Cartesian ##z## = cylindrical ##z##.
And from ##r## to ##\rho## is a breeze.

(*) you want to beware that those stupid mathematicians chose to confuse everyone by swapping ##\theta## and ##\phi##

[edit] proof of this claim: check out how wikipedia makes a mess of your exercise.
 
Last edited:
  • #5
leeban7 said:
which is the answer in spherical coordinates
Not the full answer, though ! You only mention ##E_r## :rolleyes:
 
  • #6
Due to symmetry isn't the electric field on dependant on r?
 
  • #7
The magnitude is, yes. But ##\vec E## is a vector: it also has a direction (along ##\hat r##) and you only give the radial component without stating the other two.
 
  • #8
Would I be correct in then saying that
\vec E = q/4*pi*epsilon_0*r^2 \hat r + 0 \hat omega + 0 \hat phi
 
  • #9
leeban7 said:
Would I be correct in then saying that
$$\vec E = {q\over 4\pi\varepsilon_0 r^2} \, \hat r + 0 \, \hat \phi + 0 \, \hat \theta$$
Yes, that (somewhat edited * :smile: ) is the full expression I hinted at. Hopefully makes it easier for you to convert to cylindrical.

* $$\vec E = {q\over 4\pi\varepsilon_0 r^2} \, \hat r + 0 \, \hat \phi + 0 \, \hat \theta$$
 

1. What is the formula for calculating E due to a point charge at the origin?

The formula for calculating the electric field (E) due to a point charge at the origin is E = kq/r^2, where k is the Coulomb's constant (9x10^9 Nm^2/C^2), q is the magnitude of the point charge, and r is the distance from the charge to the point where you want to calculate the electric field.

2. How is the direction of the electric field determined for a point charge at the origin?

The direction of the electric field for a point charge at the origin is determined by the direction of the electric force that would act on a positive test charge placed at that point. The electric field lines always point away from a positive charge and towards a negative charge.

3. Does the electric field due to a point charge at the origin change with distance?

Yes, the electric field due to a point charge at the origin decreases with distance according to the inverse square law. This means that as the distance from the charge increases, the strength of the electric field decreases.

4. Can the electric field due to a point charge at the origin be negative?

Yes, the electric field due to a point charge at the origin can be negative if the point charge is negative. This indicates that the direction of the electric field is towards the charge, rather than away from it.

5. How is the electric field due to multiple point charges at the origin calculated?

The electric field due to multiple point charges at the origin can be calculated by using the principle of superposition. This means that the total electric field is the vector sum of the individual electric fields due to each point charge.

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