Gauss’ Law – Summation


i’m doing some practice problems using Gauss’ law, but I feel like my work is ‘sloppy’. I’ll show an example, where I think I get the right answer, but it feels like i’m neglecting to treat the summation properly, or perhaps I don;t quite understand why what i’m doing is fine…

1. The problem statement, all variables and given/known data

A circular surface with a radius of [itex]0.072m[/itex] is exposed to a uniform external electric field of magnitude [itex]1.44[/itex]x[itex]10^{4}NC^{-1}[/itex]. The electric flux through the surface is [itex]82 Nm^{2}C^{-1}[/itex]. What is the angle between the direction of the electric field and the normal to the surface?

2. Relevant equations

3. The attempt at a solution

[itex]\Phi_{E} = \Sigma(E \cos \phi) \Delta A[/itex]

[itex]\Sigma \Delta A = \pi r^{2}[/itex]

[itex]\cos \phi = \frac{\Phi_{E}}{E \pi r^{2}} = 0.3497[/itex]

[itex]\phi = \cos^{-1}(0.3497) = 69.5 \deg[/itex]

feel like i’ve done this correctly, but I also feel like i’ve just dropped the summation sign without really knowing why. I know I summed up all the little areas to give the area of a circle, but it was also the sum of [itex]E \cos \phi[/itex].

Could some please explain why I don’t actually do a summation there?

This may be a bit of a bizarre question…


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