This problem is from Introduction to Electrodynamics 4th Edition , Griffiths, problem 3.12.
Two long, straight copper pipes, each of radius $R$, are held a distance $2d$ apart. One is at potential $V_0$, the other at $-V_0$. Find the potential everywhere.
Putting in more detail, the center of each pipes is at $(d,0)$ and $(-d,0)$. Both pipes stretch z-axis infinitely.
To solve this problem with the method of images, place infinitely two long wires at adequate points inside of each of pipes. The solution is quite tricky and complex, which is of no concern to me.
What I can't understand is why the solution obtained from the method of images can be applied to the area inside of pipes. Until now, I've assumed that you can't use the method of images to obtain the potential for the volumes which contain image charges.
To make it clear, when we deal with a problem like "Find potential everywhere when there is charge $q$ on $z$-axis and $xy$-plane is grounded to zero", we put $-q$ symmetrical to $q$ so as to xy-plane is zero. But, in this case, we can't deduce potential below the $xy$-plane. Right?
So, what makes these two problems different? Can anyone help me?
Best Answer
You will get the potential outside from the method of images. However, the potential inside is obtained from the solution of the Laplace equation inside a cylinder with the given boundary condition. (Hint: the solution is immediate.)