Why is the superposition principle valid here?

In summary, the superposition principle is valid because it allows for the analysis of complex systems by considering the effects of individual components independently. This principle holds true under conditions where interactions between components do not alter their individual behaviors, enabling simple additive solutions to be derived from the combined effects. It is particularly applicable in linear systems, where responses to multiple stimuli can be summed to predict overall behavior accurately.
  • #1
tellmesomething
410
45
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
 
Last edited:
Physics news on Phys.org
  • #2
What's the problem with your calculation? ##E(r)## is positive for ##r < R## as you have calculated it.
 
  • Like
Likes tellmesomething
  • #3
PeroK said:
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
 
  • #4
tellmesomething said:
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.
 
  • Like
Likes tellmesomething
  • #5
The center of the sphere has zero volume so there is no effective difference between the scenarios you describe.
 
  • Like
Likes tellmesomething
  • #6
Orodruin said:
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.
 
Back
Top