A Flux Divided (Part 2)

Electricity and Magnetism Level 4

Consider a closed surface consisting of a unit sphere centered on the origin in the x y z xyz coordinate system. A point-charge is placed at ( x , y , z ) = ( 1 2 , 1 2 , 1 2 ) (x,y,z) = \Big ( \frac{1}{2},\frac{1}{2},\frac{1}{2} \Big ) .

What fraction of the total electric flux through the surface passes through the upper half of the sphere (z > 0)?


The answer is 0.7876.

Steven Chase by Steven Chase , 37, USA

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1 solution

Otto Bretscher
Dec 10, 2018

By Gauss' Law, the downward flux of the given inverse square field through the southern hemisphere is equal to the downward flux through the disk D D given by x 2 + y 2 ≤ 1 , z = 0 x^2+y^2 \leq 1, z=0 , which is ∫ D 0.5 ( x − 0.5 ) 2 + ( y − 0.5 ) 2 + 0.25 d S ≈ 2.67 \int_D \frac{0.5}{(x-0.5)^2+(y-0.5)^2+0.25}dS\approx 2.67 . The fraction of the flux through the northern hemisphere is 1 − 2.67 4 π ≈ 0.788 1-\frac{2.67}{4\pi}\approx \boxed{0.788} .

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