Finding the potential between two charged conducting shells?

[JustinLiang]

Homework Statement

Suppose we have an outer shell with charge -2Q and radius b. We also have an inner shell with charge Q and radius a. what is the potential when a<r<b.

Homework Equations

E=kQ/r^2
V=kQ/r
E=-dV/dx

The Attempt at a Solution

The method I have been using according to my professor is a mathematical coincidence and the proper method to solving this is to use integrals.

How I solve it is that I sum the potential of the outer shell with the inner shell. So since we are inside the -2Q shell we have a constant V=-2Q/b. Since we are outside the Q shell we have V=Q/r.

Thus the resulting potential is V=Q/r-2Q/b.


I would like to learn the proper method where you have to take infinity as a reference and integrate. And then you get +C somewhere... lol Could someone please explain this? Thanks!

 

[anathema]

Thank you for your question. The method you have been using is indeed a mathematical coincidence and not the proper way to solve this problem. As you mentioned, the correct method involves using integrals.

To start, we can use the formula for electric potential, V=kQ/r, to calculate the potential at a point inside the inner shell (a<r<b). We can then integrate this over the entire region to find the total potential at that point.

The integral for electric potential is given by V=-∫E•dr, where E is the electric field and r is the position vector. In this case, the electric field is given by E=kQ/r^2, so we can rewrite the integral as V=-∫kQ/r^2•dr.

Since we are solving for the potential at a point between the two shells, we can set up our integral as follows: V=-∫a^b kQ/r^2•dr. Note that we are taking the limits of the integral as the inner and outer shell radii, a and b, respectively. This is because we are integrating over the entire region between the two shells.

Integrating this expression gives us V=-kQ∫a^b 1/r^2•dr. The integral of 1/r^2 is -1/r, so we can rewrite the expression as V=kQ(1/a-1/b). This is the proper way to solve for the potential at a point between two charged shells.

I hope this explanation helps you understand the proper method for solving this problem. If you have any further questions, please do not hesitate to ask. Thank you for your interest in science and for striving to learn the correct methods for solving scientific problems. Keep up the good work!