Calculating Electric Field at the Center of a Hemispherical Shell of Charge

[swervin09]

Homework Statement

A hollow sphere of outer radius R2 and inner radius of R1 carries a uniform charge 2Q. The sphere is then cut in half to create a hemispherical shell of charge Q. Calculate E at the center point (origin) P.

Homework Equations

equation of a hollow sphere = 2/3π(r2-r1)
Gauss' Law ∫E dot dA
surface area hemisphere = 2πr^2

The Attempt at a Solution

Well, I know this is an integration problem and that I am better off integrating with polar coordinates and that I will be integrating from 0-->π as my lower and upper integral bounds.
But in all honesty I haven't had much fortune setting the integral up. The set up is the help I am asking for.

 

[serllus reuel]

If I am not mistaken, you need to find the E field due to a HEMISPHERICAL shell

That involves some somewhat-complicated multiple integrals and E form Coulumb's law. Gauss's law will not work due to lack of symmetry

 

[serllus reuel]

Assuming that that is your problem, here is my hint:

Use spherical coordinates. find the charge density. consider an infinitesimal piece of the shell and the Coulomb force on a test charge at the origin. Then choose appropriate limits for r, theta, and phi and integrate

 

[swervin09]

Yes that is the problem. spherical coordinates make more sense. I will try that and post tomorrow what I have come up with. I don't have my calculus book nearby to refresh my memory of spherical coord. integration. This is one of those problems that has me intrigued and eager to "beat". It isn't worth a lot of points but it is due Thursday.
Thank you for the hint!

 

[serllus reuel]

no problem!

I might add: depending on how comfortable you are with multivariable, you don't have to use a triple integral; just find the field of a ring, sum into a washer, sum the washers into a shell. The spherical coordinates are just a way of thinking, no need to get formal about it

 

[swervin09]

I follow you up until you state sum the washers into a shell. Please provide a hint as to that specific.