Finding Potential due to hemispherical charge distribution.

[cat_facts]

Homework Statement

Given that the hemisphere has a uniform charge density σ distributed through its inner surface with radius a. Find the electric potential at the base of the hemisphere and the top of the hemisphere (that point where the radius is the largest).

Homework Equations

V = ∫ (kσ / [r-r'])*da (Surface Integral)

The Attempt at a Solution

I tried seperating this hemisphere into rings integrating from of the bottom of the hemisphere to the top, but that isn't working. I was trying to then solve it for all z along z(hat), then plugging in 0 and the height of the hemisphere, but wasn't sure how to proceed.

I tried V = ∫ (kσ / [r-r'])*da = kσ∫ (1/ [r-r'])*da = kσ∫ (1/ [h^2 + a^2])*2pi*radius*dh = 2*pi*kσ∫ (a/ [h^2 + a^2])dh

but know I'm not proceeding correctly. What should I do?

 

[mark726]

Hello!

To find the electric potential at the base and top of the hemisphere, we can use the formula for electric potential due to a charged hemisphere, which is V = (2/3) * k * σ * a, where k is the Coulomb's constant, σ is the charge density, and a is the radius of the hemisphere.

At the base of the hemisphere, the radius is equal to a, so the potential becomes V = (2/3) * k * σ * a.

At the top of the hemisphere, the radius is equal to 2a, so the potential becomes V = (2/3) * k * σ * 2a = (4/3) * k * σ * a.

So, the electric potential at the base of the hemisphere is (2/3) * k * σ * a and at the top of the hemisphere is (4/3) * k * σ * a.

Hope this helps! Let me know if you have any further questions.