(adsbygoogle = window.adsbygoogle || []).push({}); There is a hemisphere of radius R and surface charge density [itex]\sigma[/itex]. Find the electric potential and the magnitude of the electric field at the center of the hemisphere. I started by saying [tex]V = \int dV = \int \frac{\sigma dA}{4\pi \epsilon_0 R}[/tex]. This, at least, I am confident is correct.

Then, I changed [itex]dA[/itex] into [itex]Rd\theta[/itex] and [itex]Rd\alpha[/itex], where the first one goes across the hemisphere's surface, and the second one goes around the hemisphere's circular edge.

Continuing:

[tex]

V = \frac{\sigma R}{4\pi \epsilon_0}\int_{0}^{\pi/2}d\theta \int_{0}^{2\pi}d\alpha[/tex]

Which, when evaluated, gives me the wrong answer.

Oblivious to my error at the time, I continued to the second part.

[tex]E = \int dE = \int \frac{\sigma dA}{4\pi \epsilon_0 R^2}\hat{r}[/tex].

I said that [tex]\hat{r} = \cos{\theta}\hat{i} + \sin{\theta}\hat{j}[/tex]. Using similar logic as I did for the potential part of the problem, I continued:

[tex]

E = \frac{\sigma R}{4\pi \epsilon_0}\int_{0}^{\pi/2}\cos{\theta}\hat{i} + \sin{\theta}\hat{j} d\theta \int_{0}^{2\pi}d\alpha[/tex]

Which, when evaluated, also gives me the wrong answer. The correct answers for the parts are, respectively, [tex]V = \frac{\sigma R}{2 \epsilon_0}[/tex] and [tex]E = \frac{\sigma}{4\epsilon_0}[/tex].

I have no idea what I'm doing incorrectly. I've never done a three-dimensional integration before, but this is the way that I thought it should be done. What have I done wrong?

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# Homework Help: Electric Potential Hemisphere Problem

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