Can the Sphere of Radius 'a' Be Modeled as a Point Particle?

[Abwi]

Homework Statement

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Find the electric force between the two spheres.
Sphere r=b-a has a volumetric density of p=K, where K is a constant
Sphere r=a has a volumetric density of p=θ*r

Homework Equations

As you can see the sphere of radius 'a' doesn't have a uniform electric field because it varies with respect to 'θ' and 'r'. I need to know if the sphere of radius 'a' can be modeled as a point particle even if isn't uniformly charged?

The Attempt at a Solution

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The upper sphere can be modeled as a point particle because it is uniformly charged so I can obtain its charge which is Q =

For the sphere centered at the origin I define an r vector between a point P inside the sphere of radius a in respect to the center of the upper sphere. Which after some arrangements leaves me with this expression for it's electric displacement field D =

This field goes on the positive z direction.

And this integral cannot be solved analytically as far as I know. Which is why I want to know if that sphere can be modeled as a point particle so I can just multiply the charges of each sphere?

 

[Buzz Bloom]

I need to know if the sphere of radius 'a' can be modeled as a point particle even if isn't uniformly charged?

Hi Abwi:

It has been a very long time since I was into this topic, but I am pretty sure the answer to the above question is NO. It may be possible for a non-uniform charge distribution over a spherical surface to be equivalent to all the charge at the center, but in such case the distribution would have to have certain symmetries. However since the upper sphere has uniform charge, its force w/r/t other charged sources is the same as if it were a point charge at its center. If the lower sphere's distribution is symmetric w/r/t the line connecting the two centers, the the lower sphere's force on the upper sphere would be the same as if it were a point charge at some point on this line, but not necessarily the center of the lower sphere.

Hope this helps.

Regards,
Buzz

 

[rude man]

Is θ the polar or azimuth angle (latitude or longitude)?