Finding electric flux across a hemisphere?

Question

Let M be the closed surface that consists of the hemisphere:

M1: x^2+y^2+z^2=1,  z≥0
and its base:
M2: x^2+y^2≤1     z=0

Let E be the electric field defined by E=(3x,3y,3z). Find the electric flux across M. Write the integral over the hemisphere using spherical coordinates, and use the outward pointing normal.

∫∫M1  E·dS=  a∫b c∫d f(θ,φ) dθdφ

So, I have to find a, b, c, and d (I'm assuming the bounds for theta are 0 2pi and for phi they're 0 to pi/2, since it is a hemisphere).

I also have to find f(θ,φ) and the integrals ∫∫E·dS over M1, M2, and M.

I'm pretty lost here, any advice?

kb · Accepted Answer

Doing this with the divergence theorem we can avoid direct evaluation of the surface integral(s).
∫∫m E · dS
= ∫∫∫ (div E) dV
= ∫∫∫ (3 + 3 + 3) dV
= 9 * (volume of unit hemisphere)
= 9 * (1/2)(4π/3)
= 6π.

I hope this helps!

Finding electric flux across a hemisphere?

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