B Method of images and spherical coordinates

AI Thread Summary
The discussion revolves around calculating the potential due to a point charge and its image charge using spherical coordinates. The user initially struggles with the expression for the distance between the real charge and the image charge, particularly regarding the terms involving the angle θ. They clarify that the angle θ should be considered as the angle between the vectors r and R', leading to confusion about whether to use the dot product or a simple product in the potential formula. Ultimately, they realize that the correct interpretation of the angles resolves their issue, confirming that the potential is indeed zero on the xy-plane as expected. The conversation emphasizes the importance of accurately defining angles in spherical coordinates for potential calculations.
josephsanders
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I am having trouble with the magnitude of distance vector in spherical coordinates.
I am finding the potential everywhere in space due to a point charge a distance 'a' on the z-axis above an infinite xy-plane held at zero potential. This problem is fairly straight forward; place an image charge q' = -q at position -a on the z-axis. I have the solution in cartesian coordinates but I am having trouble writing it in spherical coordinates and this bothers me.

Suppose the position vector of the real charge is R = a\hat{z} and the image charge is R'= -a\hat{z}. So in spherical coordinates the magnitude of |r-R'| = (r^2 + R'^2 - 2rR'cosθ)^(1/2). My question lies in the last term. If "2rR'" refers to the magnitude of r times the magnitude of R' that would imply that the potential is zero everywhere since |R'| = |R|. So then I thought maybe it would make more sense to write |r-R'| = (r^2 + R'^2 - 2r⋅R'cosθ)^(1/2). Because then we would get that the potential is only zero on the xy-plane as desired. However when we are deriving the distance formula in spherical coordinates from the cartesian coordinates we have that (x,y,z) = (rsinθcosφ,rsinθsinφ,rcosθ) where the 'r' here does refer to the magnitude so I don't see how we can get the dot product that we would need to have the potential not vanish.

I hope I explained what I was confused on well enough! I know that this is the right answer for the method of images problem with a conducting sphere so I must be thinking about something wrong.EDIT: Actually I understand my error now. I wasn't thinking of θ as the angle between R' and r. So the angle in the two magnitude calculations is not actually the same angle. Thank you to those who helped!
 
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The difference in the two formula you show is 2r⋅R'cosθ or 2rR'cosθ. They have a same value.
 
anuttarasammyak said:
The difference in the two formula you show is 2r⋅R'cosθ or 2rR'cosθ. They have a same value.
I don't think they are the same since if we use the dot product definition r⋅R' = rR'cosθ, then that would mean 2r⋅R'cosθ = 2rR'cos^2θ right?
 
I see you mean by r⋅R' inner product of vectors
\mathbf{r} \cdot \mathbf{R'}
Then mathematics says that side length of triangle with other side lengths r,R' with angle ##\theta## between is
|\mathbf{r}-\mathbf{R'}|=\sqrt{r^2+R'^2-2rR'\cos\theta}
On the x-y plane where ##\theta=\pm \pi/2##, ##\cos \theta=0## which gives zero potential there as you expect.
 
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anuttarasammyak said:
I see you mean by r⋅R' inner product of vectors
\mathbf{r} \cdot \mathbf{R'}
Then mathematics says that side length of triangle with other side lengths r,R' with angle ##\theta## between is
|\mathbf{r}-\mathbf{R'}|=\sqrt{r^2+R'^2-2rR'\cos\theta}
On the x-y plane where ##\theta=\pm \pi/2##, ##\cos \theta=0## which gives zero potential there as you expect.
@josephsanders ,

In terms of the angle, ##\theta##, as you are using it as a spherical coordinate, the angle between ##\mathbf{r}## and ## \mathbf{R'}## is ##\pi - \theta##. The angle ##\theta## is between the positive z-axis and ##\mathbf{r}##, where as ## \mathbf{R'}## lies along the negative z-axis.

Since ##\cos (\theta) = -\cos(\pi - \theta)## the result you need is

##\displaystyle \quad \quad |\mathbf{r}-\mathbf{R'}|=\sqrt{r^2+R'^2+2rR'\cos\theta \ }##
 
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