Triple integral in eletromagnetism: prove force doesnt depend radius.

[tsuwal]

Homework Statement

This problem may be dull, I know, but maybe there is a hidden math trick that i don't know of. This picture sums up the problem.

So, you should prove by simplifing the integral that $F^e$, the eletric force applied between two spheres, onde with a charge $q_1$ and the other with the charge $q_2$ (distributed evenly in volume, with a charge density $\rho$) doesn't depend on $R_2$, it only depends on $q_1, q_2$ and $d$

Homework Equations

$F^{e}=\int_V \frac{q_1\rho }{4\pi \delta^{2}} cos(\phi ) dV$

The Attempt at a Solution

$F^{e}=\int_V \frac{q_1\rho }{4\pi \delta^{2}} cos(\phi ) dV =\frac{q_1q_2 }{4\pi\frac{4}{3}\pi R_2^{3}}\iint_{0}^{R_2}\frac{cos(\phi )sin(\phi )r^2}{\delta^{2}}drd\phi=\frac{q_1q_2 }{4\pi\frac{4}{3}\pi R_2^{3}}\iint_{0}^{R_2}\frac{cos(\phi )sin(\phi )r^2}{(d-rcos(\phi )^2+(rsin(\phi )^2))} drd\phi=\frac{q_1q_2 }{4\pi\frac{4}{3}\pi R_2^{3}}\iint_{0}^{R_2}\frac{cos(\phi )sin(\phi )r^2}{(d^2+r^2-2drcos(\phi ))} drd\phi$

How do you simplify this integral or at least show that the expression doesn't depend on $R_2$? I tried to derivate with respect to $R_2$ but it didn't helped...

 

[mfb]

If I remember correctly, it is easier to integrate over a spherical shell only. If this is equivalent to a point-charge, the remaining part is easy.

Do you have to use integration? Gauß' law would be so nice...

 

[tsuwal]

You are saying that first we could show that a sphere with some charge density is equal to a spherical surface with a surface density choosen to have the same total charge and then simplify the integral. Yeah that could work but still get a pretty dull integral. I just wanted to know if there was some hidden math trick but seems like Gauss's law is the way to go.

 

[mfb]

No, the idea is to get rid of the r-integration: Show that a spherical shell has the same potential (at some specific point) as a point-charge in the center of the shell. The filled sphere just consists of "many" spherical shells.

 

[tsuwal]

Show that a spherical shell has the same potential.

yeah but i was asking for the eletric force not the potencial...

 

[tsuwal]

I got the many shperical shells=sphere with some volume density though

 

[mfb]

yeah but i was asking for the eletric force not the potencial...

If the potential is the same, the electric field and the force are the same as well.

Oh, I think I see the problem: The force is a vector (with variable orientation), you have to add vectors and not their magnitude. That gets messy - calculate the potential instead, this should be easier.