Use Gauss' Law for Electricity to find the total charge?

Question

Use Gauss' Law for Electricity to find the total charge in the hemisphere z=sqrt(R^2-x^2-y^2) for E=<4x-y, 2y+z, 3xy>. (Answer: 4pi*Eo*R^3)

The divergence is 6, I've set up the triple integral of 6 dV. But how do I find the radius of this hemisphere?

Answer 1

The radius of the hemisphere is R, which is given.

(Attn: Square both sides of z = sqrt(R^2-x^2-y^2) and simplify: x^2+y^2+z^2 = R^2)

∫∫E⋅dA = ∫∫∫div E dv = ∫∫∫6 dv = (1/ε_o)(total charge)

So, the total charge inside the hemisphere = ε_o*6 (2/3)pi R³ = 4pi ε_o R³