Is Gauss' Law Summation Necessary for Uniform Electric Fields?

[BOAS]

Hello,

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...

Homework Statement

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

Homework Equations

The Attempt at a Solution

$\Phi_{E} = \Sigma(E \cos \phi) \Delta A$

$\Sigma \Delta A = \pi r^{2}$

$\cos \phi = \frac{\Phi_{E}}{E \pi r^{2}} = 0.3497$

$\phi = \cos^{-1}(0.3497) = 69.5 \deg$

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 $E \cos \phi$.

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

This may be a bit of a bizarre question...

Thanks!

 

[Chestermiller]

Hello,

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...

Homework Statement

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

Homework Equations

The Attempt at a Solution

$\Phi_{E} = \Sigma(E \cos \phi) \Delta A$

$\Sigma \Delta A = \pi r^{2}$

$\cos \phi = \frac{\Phi_{E}}{E \pi r^{2}} = 0.3497$

$\phi = \cos^{-1}(0.3497) = 69.5 \deg$

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 $E \cos \phi$.

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

This may be a bit of a bizarre question...

Thanks!

You've done it correctly. $E$ and $\cos \phi$ are constant over the area because the electric field is uniform, so they can come out of the summation.

Chet