Find average potentail over a sphere

[leonne]

Homework Statement

Find avg potentail over a spherical surface of radius R due to a point charge q inside. show that in general vave = vcenter + Qenc/4pieEoR

Homework Equations

The Attempt at a Solution

I already got the answer but review for final and forgot some stuff.
1) I am going to use the formula V(r)=1/4pieR² enclosed integral Vda
2) I find what V is which i got V=1/4pieE0(q/r) My question is how do i find this V? looking at my notice i see volume charge V=1/4pieEo integral P(r)/r dt do I use this? If so what is p(r)? I am guessing its q but why?

Than I have r²= z² +R² -2zRcos@ why is this? Or is this some formula?
3) Than i plug all that into the formula and solve it which i understand how.
Thanks

 

[mark726]

for any help!

Hello,

To find the potential over a spherical surface of radius R due to a point charge q inside, we can use the formula V(r) = 1/4πε₀∫ρ(r')/|r-r'| dτ, where ρ(r') is the volume charge density and dτ is the volume element. In this case, we are dealing with a point charge, so the volume charge density is simply q. Therefore, we can rewrite the formula as V(r) = 1/4πε₀∫q/|r-r'| dτ.

Next, we need to find the expression for |r-r'|, which represents the distance between the charge at the center (r') and the point on the spherical surface (r). This can be found using the Pythagorean theorem, where |r-r'| = √(z²+R²-2zRcosθ). This formula represents the distance between two points in spherical coordinates, where z is the distance along the z-axis and θ is the angle between the z-axis and the line connecting the two points. This formula is derived from the Law of Cosines, which is a general formula for finding the distance between two points in any coordinate system.

Finally, we can plug this expression into our original formula for V(r) and integrate over the entire surface of the sphere, which gives us the average potential over the spherical surface. This average potential can be shown to be equal to the potential at the center of the sphere (vcenter) plus the potential due to the enclosed charge (Qenc/4πε₀R). This is because the potential at the center is simply the potential due to the point charge, while the potential due to the enclosed charge is the average potential over the entire surface. Therefore, we have vave = vcenter + Qenc/4πε₀R, as desired.

I hope this helps clarify any confusion you may have had. Let me know if you have any further questions. Good luck on your final exam!