Electric field due to 2 charged spheres

[Yoni V]

Homework Statement

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.

Homework Equations

E=E1+E2

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

 

[Orodruin]

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.

 

[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)

 

[Orodruin]

Are you familiar with series expansions?

 

[Yoni V]

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

 

[Orodruin]

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.

 

[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...

 

[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!