Gauss's Law - Ring of Charge

A ring of charge is defined as follows:

x = cos θ y = sin θ z = 1 0 θ 2 π x = \cos \theta \\ y = \sin \theta \\ z = 1 \\ 0 \leq \theta \leq 2 \pi

The ring has + 1 +1 units of charge per unit length. Let ϕ 1 \phi_1 be the electric flux through the + z +z half of a sphere of radius 2 2 centered on the origin, and let ϕ 2 \phi_2 be the electric flux through the z -z half.

Determine the following ratio:

ϕ 1 ϕ 2 ϕ 1 + ϕ 2 \frac{\phi_1 \, \phi_2}{\phi_1 + \phi_2}

Details and Assumptions:
1) Electric permittivity ϵ 0 = 1 \epsilon_0 = 1
2) Use outward-facing surface normal vectors

Note: I probably should have rated this "Hard", but I clicked "Medium" instead


The answer is 1.17.

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