# Electric field at center of charged hemispherical shell

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1. Nov 1, 2015

### Titan97

1. The problem statement, all variables and given/known data
Find Electric field at center of charged hemispherical shell

2. Relevant equations
In spherical coordinates, $$dA=R^2\sin\phi d\phi d\theta$$

3. The attempt at a solution

From the image, Enet=2dEy=2dEcosφ
$$dE=\frac{kdq}{R^2}E$$
$$dE=\frac{k\sigma dA}{R^2}$$
$$dE=\frac{k\sigma R^2\sin\phi d\phi d\theta }{R^2}$$
$$dE_{total}=2k\sigma\sin\phi\cos\phi d\phi d\theta$$
$$E=\int_0^{2\pi}\int_0^{\frac{\pi}{2}}2k\sigma\sin\phi\cos\phi d\phi d\theta$$
$$E=\frac{\sigma}{2\epsilon_0}$$
But given answer is $$E=\frac{\sigma}{4\epsilon_0}$$

2. Nov 1, 2015

### blue_leaf77

The integration over the azimuthal angle from 0 to $2\pi$ already takes the factor of two in your expression $dE_{total}=2k\sigma\sin\phi\cos\phi d\phi d\theta$ into account. So you should have removed it from $dE_{total}$.

3. Nov 1, 2015

Thank you.