Why is the superposition principle valid here?

[tellmesomething]

Homework Statement

An early model for an atom considered it to have a positively charged points nucleus of charge Ze, surrounded by a uniform density if negative charge up to radius R. The atom as a whole is neutrality. For this model, what is the Electric field at a distance r from the nucleus

Relevant Equations

None

This is a discussion for (r<R).

Assuming a gaussian surface at x=r from the center we get

$$E(r) = \frac{Ze} {4π\epsilon_0} ( \frac{1} {r²} - \frac{r} {R^3} )$$

However we get the same result if we consider a wholly negatively charged solid sphere and find the field at a distance r inside the sphere and add it with the field due to a single point charge kept at the centre of such a sphere...

$$E(r)=E_{-ve sphere}+E_{+ve point charge}$$

How can we consider the field due to the whole negative sphere, isnt the middle albeit being a very small point charge positive instead of negative?

field due to a wholly negatively charged sphere + field due to a point positive charge≠ Field due to a negatively charged sphere with its midpoint being empty+ field due to a point charge

Is this just an approximation? Or do I not know how to apply the superposition principle. Please consider helping out

 

[PeroK]

What's the problem with your calculation? $E(r)$ is positive for $r < R$ as you have calculated it.

 

[tellmesomething]

What's the problem with your calculation? $E(r)$ is positive for $r < R$ as you have calculated it.

Thats not the problem. The problem is that it matches the field of a wholly negatively charged sphere and a positive point charge on it. But in this case it isnt a wholly negative charged sphere the mid point has a positive charge. So how can it be considered?

Thats how we apply superposition principle right?

We consider the fields due to both the distributions

So field due to a wholly negatively charged sphere + field due to a point positive charge≠ Field due to a negatively charged sphere with its midpoint being empty+ field due to a point charge

 

[PeroK]

Thats not the problem. The problem is that it matches the field of a wholly negatively charged sphere and a positive point charge on it. But in this case it isnt a wholly negative charged sphere the mid point has a positive charge. So how can it be considered?

Thats how we apply superposition principle right?

We consider the fields due to both the distributions

So field due to a wholly negatively charged sphere + field due to a point positive charge≠ Field due to a negatively charged sphere with its midpoint being empty+ field due to a point charge

I don't understand this. It matches a positively charged solid sphere.

 

[Orodruin]

The center of the sphere has zero volume so there is no effective difference between the scenarios you describe.

 

[tellmesomething]

The center of the sphere has zero volume so there is no effective difference between the scenarios you describe.

Oh. That makes sense now. Thankyou so much.