# Calculating electric field at an uncharged portion in an otherwise uniformly charged sphere

We were doing some problems on electric fields and my teacher discussed this one :

Prerequisite: The electric field at a radial distance r inside a uniformly charged sphere of charge density ρ is given by $$\frac{ρ\vec r}{3ε_o}$$

Q) There are two oppositely charged spheres of uniform charge density ρ(pink) and -ρ(green). We fuse them together so that the vector joining O to O' is given by $$\vec a$$. Find the electric field at any point P inside common region .

Solution) Due to being oppositely charged, the common fused portion becomes overall uncharged. We calculate the electric field at P(in figure) due to left and right spheres individually. $$\vec E_O=\frac{ρ\overrightarrow r}{3ε_o}$$ $$\overrightarrow E_{O'}=\frac{-ρ\overrightarrow {r'}}{3ε_o}$$ Since : $$\overrightarrow a+\overrightarrow {r'}=\overrightarrow r$$ We can conclude: $$\overrightarrow E_O+\overrightarrow E_{O'}=\frac{ρ\overrightarrow {a}}{3ε_o}$$

I had a doubt that why are we using a relation derived for uniformly charged sphere for the uncharged portion in which P is located. The point P is not inside charged portion of the sphere, so the formula should not be applicable at that point. I hope you guys understand that.

I asked this to my teacher and his reply was "Result itself is the explanation" . Am I missing something here ? And how can we explain a result whose explanation is the result itself ?

I have come here as a last resort hoping that I could get a satisfactory answer.

PS: This is not a homework question or off-topic. I don't understand a conceptual thing, so moderators please have mercy on me.

• As a note: I agree this isn't an off topic question, but typically question closure comes from regular users, not moderators. – BioPhysicist Apr 29 '20 at 17:22

• Suppose I removed one of the spheres now, since the material isn't conducting, the uncharged portion would remain uncharged. Then, would the electric field at point P still be the same ? P is not on the inner surface of that uncharged arc , if it was, then i understand that the field would still be given by $\frac{{\rho}\overrightarrow r }{3e}$.In this scenario, I think it would change. Now bring back the other sphere. I hope you get the point I'm trying to make. – Physicsa Apr 29 '20 at 17:57
• @Physicsa I don't understand. If you take one of the spheres away then $P$ is inside of the other sphere, so $P$ is now in a location with charge. With only one sphere the field is described by the equation you have given for the field due to one sphere. – BioPhysicist Apr 29 '20 at 18:05