Manually integrating to find flux through hemisphere

[yango_17]

Homework Statement

Basically, I am being asked to calculate the electric flux through the top of a hemisphere centered on the z-axis using "brute force integration" of the surface area.

Homework Equations

Gauss' Law

The Attempt at a Solution

Using intuition and Gauss' law, I know that the flux is Φ=Eπr^2, but I'm not sure how to prove this using "brute force" integration as my teacher requires. If anyone could help me at least set up the integral, it would be much appreciated.

 

[BiGyElLoWhAt]

I'm not sure what you mean by "brute force". It can be readily shown that the projection of a sphere onto theal plane of said sphere is pi*r^2, and since phi=E(dot)dA or just E(dot)A for approximation, then you end up with that expression. Perhaps show that E(dot)A is E*pir*r^2

 

[yango_17]

I'm not sure exactly what he means either haha. Anyway, he said it'd be easier to do if we use the spherical coordinate system to represent the hemisphere, which sounds like a hint, but I'm still not too sure how to proceed.

 

[BiGyElLoWhAt]

Hmmm... I'm not sure I agree with that, but it can definitely be done in spherical coordinates. You are familiar with the equation of a sphere, no?

Edit* Wait, I lied. It makes the limits way nicer.

 

[yango_17]

No, I'm afraid I don't know how to represent a hemisphere in spherical coordinates

 

[BiGyElLoWhAt]

I see. Well a sphere takes the form $<r,\phi,\theta>$. That is an arbitrary point on the surface of a sphere, and you need to find the normal to the tangent plane at each point, then dot it with the electric field, and integrate over the surface.
$\Phi = \int_S \vec{E} \cdot d\vec{A}$

 

[BiGyElLoWhAt]

By brute force, he might have just meant make the long calculation.

 

[yango_17]

Alright, I'll see if I can work it out from here. Thanks!

 

[BiGyElLoWhAt]

No problemo!