I don't understand an assumption the textbook made ?

[twotaileddemon]

Homework Statement

It comes from example 3.2 in griffith's 3rd edition electrodynamics book

A point charge q is situated a distance a from the center of a grounded conducting sphere of radius R. Find the potential outside the sphere.

Homework Equations

The Attempt at a Solution

I can follow the example, but it assumes that there is another point charge q' = -(R/a) * q, where q' is a distance b from the center (b = R^2/a) and q is a distance a from the center or a distance a-b from q'

My question is... where does it get such an assumption? Or, how does one know to chose such an arbitrary expression? I'm mostly interested in the relationship between the two charges... and how the charge q' (on the inside) is determined.

Thanks for your help

 

[CompuChip]

Actually Griffiths does explain that, at the bottom of that page (the paragraph below (3.18)). You can try the method he explains: take any charge q' at any position x' and calculate the resulting potential, and see which q' and x' to choose to recover (3.17) or (3.19).

Usually, if the location of the image charge is not obvious from the problem, a hint will be given. If you have any arbitrary problem, first try with a single image charge, and check if it does the job. If not, you can go further but probably the method of images isn't such a good idea at all.

 

[twotaileddemon]

So I should calculate the potential between q and q'...

ok, I get that.
But even if the potential is given, if I integrated using the equation V = -I(E.dl) where I is integral E is the electric field and . is the dot product, would I get the same result for V?

and once I get the potential.. since the sphere is grounded.. I can set it equal to zero and solve for q' in terms of q?

 

[CompuChip]

I don't really get what you are asking, sorry.
You have one point charge given, and the potential at the boundaries of some area of space (namely, at the surface of the sphere and at infinity). The idea is that you replace the sphere by one (or more) charge(s) outside that region (in this case, inside the sphere) in such a way that the potential on the boundary and charge distributions inside the region are unchanged. Then the potential and derived quantities (such as the electric field) are the same in the original and new case.

So in this case, you would replace the sphere with a charge q' somewhere inside the region where the sphere was, and calculate the position and magnitude of q' such that the potentials of q and q' together vanish at distance R from the origin.

 

[twotaileddemon]

okay, I understand it now. thanks very much :)