# Field lines near the equilibrium point

1. Feb 21, 2014

### scotshocker

1. The problem statement, all variables and given/known data

Charges 4q and -q are located at the points (-2a,0,0) and (-a,0,0), respectively. Write down the potential θ(x,y) for points in the xy plane, and then use a Taylor expansion to find an approximate expression for θ near the origin, which you can quickly show is the equilibrium point. (You can set a=1 to make things simpler)

2. Relevant equations
θ(x,y,s)=∫(ρ(x',y',z')dx'dy'dz')/4πεr

3. The attempt at a solution
setting a-1, ignoring the factor of q/4πε0, the potential due to the two charges, at locations in the xy plane is θ(x,y)=(1/(sqrt((x-2)2+y2)-(1/sqrt((x-1)2+y2)

Have I set this equation up correctly?

2. Feb 21, 2014

### Simon Bridge

Welcome to PF;
... how would you check?

i.e. is the equilibrium point at the origin as the problem says?
(Notice that one of the charges is much bigger than the other? Have you accounted for that?)

Have you checked that you have added the vectors properly - say, by sketching them out head-to-tail?

Last edited: Feb 21, 2014
3. Feb 21, 2014

### TSny

Hello, scotshocker. Welcome to PF!

Looks good except for a couple of signs. At what values of x would you expect the potential to be undefined?
[EDIT: And as Simon points out, the charges are not of equal magnitude.]

Last edited: Feb 21, 2014
4. Feb 22, 2014

### scotshocker

The fact that the one charge is larger than the other is the part that I am having trouble with. I am also confusing myself because both charges are on the same side of the origin. I am severely out of practice and am just trying to figure this out. Any suggestions?

5. Feb 22, 2014

### Simon Bridge

The charges are also opposite signs.
Perhaps you'd better write down the equation for the potential along just the x axis to start with - don't leave off the q and the a this time: I think you removed them too soon, before you understood the situation.
Once you see that, you'll probably make the connections you need.

Actually draw the axes - mark out x=0, x=+a, x=+2a, x=-a, x=-2a, etc.