- #1
MarcZero
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Homework Statement
A sphere of radius a has a charge density which varies with distance r from the center according to http://img14.imageshack.us/img14/9577/wangsnessproblem594.gif where A is a constant and http://img35.imageshack.us/img35/555/wangsnessproblem593.gif .[/URL] Find the scalar potential http://img35.imageshack.us/img35/9326/phi2.gif at all points inside and outside the sphere by using the following formula:
http://img17.imageshack.us/img17/4628/wangsnesseqn572.gif
Express your results in terms of the total charge Q of the sphere.
Homework Equations
Volume of the Sphere: http://img10.imageshack.us/img10/8776/spherevolume.gif
Vector Capital R: (From Source to Point) http://img21.imageshack.us/img21/2586/captialr.gif
The Attempt at a Solution
I'm not sure if I'm over-simplifying this but I figured since the charge density is directly related and varies according to the volume, that is, that I can think of the sphere as containing a series of spheres layered on each other that are infinitely thin, I could take the integral and relate Q to the charge density and the volume. So:
http://img197.imageshack.us/img197/5717/59work1.gif
Then:
http://img197.imageshack.us/img197/3756/59work2.gif
Finding that:
http://img197.imageshack.us/img197/5041/59work3.gif
Plugging that into the main formula above, I get:
http://img197.imageshack.us/img197/9933/59work4.gif
And putting that in terms of Q:
http://img180.imageshack.us/img180/7451/59work5.gif
So if I am reading the question correctly, this is the final formula it asks for. I am really suspicious that I missed something and am forgetting a concept that will make this problem more complex than I think it is. I might also be over-thinking things as well. Any insight would be appreciated. Thank you for your time in advance and I apologize about the pictures, as I am still learning MathType.
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