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Finding electric flux across a hemisphere?
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?
I need to show all the steps though, can you walk me through the surface integrals?
1 Answer
- kbLv 79 years agoFavorite 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!