Electric field at center of charged hemispherical shell

[Titan97]

Homework Statement

Find Electric field at center of charged hemispherical shell

Homework Equations

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

The Attempt at a Solution

From the image, E_net=2dE_y=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 $$E=\frac{\sigma}{4\epsilon_0}$$

 

[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 $dE_{total}$.

 

[Titan97]

Thank you.