Homework Help: Electric field due to 2 charged spheres

1. Apr 12, 2015

Yoni V

1. The problem statement, all variables and given/known data
The first part is to calculate the electric field everywhere in space given a body of 2 spheres of radius R and distance d apart (d<R), located on the z-axis, with charge density ρ and -ρ.
Of course when r>>R this is essentially a dipole.
The second part is to approximate the field outside the body given R>>d, i.e. the 2 spheres almost entirely overlap.

2. Relevant equations
E=E1+E2

3. The attempt at a solution
Using the superposition principle I got to the following expression for the electric field outside the body (before the approx.):
E=kρ4/3piR^3[(r-d/2z-hat)/|r-d/2z-hat|^3-(r+d/2z-hat)/|r+d/2z-hat|^3]
(sorry, couldn't get the latex to work...)

Now, if R>>d I approximated it to be zero:
E=kρ4/3piR^3[r/|r|^3-r/|r|^3]=0

It kinda makes sense as a crude approximation because the charges almost cancel entirely. But I'm pretty sure this not how I'm expected to approximate it. I'm guessing some sort of leading order term, but I don't know how to pick it out from the above expression.

Thanks!
Johnathan

2. Apr 12, 2015

Orodruin

Staff Emeritus
Naturally, the leading terms for each of the potentials cancel. However, in order to get a non-zero contribution, you need to look to the next order (linear) in the quantity d/R rather than simply letting d->0.

3. Apr 12, 2015

Yoni V

Ok I think I understand, but I'm not sure how to handle the 3rd power in the norm of the vector.
If it were a 2nd power, then defining x=d/r I would get something like (simplifying a little for readability):
1/|r[rhat]+dz[zhat]|^2=1/[r^2(1+2xz/r+x^2)]
which could then be approximated as
(1/r^2)*(1-2xz/r-x^2)

But in the original case, I'm not sure how to treat the vectors. Is something like the following makes sense?
|r-d/2z-hat|^3=|r^3[rhat]-3/2dzr^2[zhat]-3/2d^2z^2r[rhat]+d^3z^3/8[zhat]|

Or could it just be
|r-d/2z-hat|^3=(r^3-3/2dzr^2-3/2d^2z^2r+d^3z^3/8)

4. Apr 12, 2015

Orodruin

Staff Emeritus
Are you familiar with series expansions?

5. Apr 12, 2015

Yoni V

We just started the course and have yet to cover series expansions...

6. Apr 12, 2015

Orodruin

Staff Emeritus
Just to be clear, I mean series expansions such as Taylor series expansions, not multipole expansions, which I suspect is something you would cover later.

7. Apr 12, 2015

Yoni V

Not very thoroughly, but I'm familiar with common expansions such as sin/cos/e^x/ln(x), and with the idea of differentiating and dividing by the nth power etc.
I think I'm having more trouble with dealing with vector side of the problem, or maybe I'm just missing something....

8. Apr 12, 2015

Yoni V

Ok I got it... It was just a matter of expressing r in cartesian coordinates, and transforming the absolute value in terms of root of square...

Thank you!