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RedDelicious

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## Homework Statement

edit: I had put this in the calculus section because it was a problem from Stewart but I guess it's closer to a physics problem considering the use of Gauss's Law. My apologies for any confusion this my

[/B]

I've been trying to do this problem without making use of the divergence theorem but have been doing something wrong and cannot figure it out.

Use Gauss’s Law to find the charge contained in the solid hemisphere

[tex]x^2+y^2+z^2\le a^2,\:z\ge 0[/tex]

if the electric field is

[tex]E\:=\:<x,\:y,\:2z>[/tex]

## The Attempt at a Solution

[/B]

I use the parametric equation for a sphere to model the hemisphere, with the idea that only the limits of integration over the parameter domain at the end will change, but the parameterization is the same.

[tex]r\left(\phi ,\theta \right)=\:<asin\phi cos\theta ,\:asin\phi sin\theta ,\:acos\phi >[/tex]

and using this to parameterize the electric field for the hemisphere

[tex]F=\:<asin\phi cos\theta ,\:asin\phi sin\theta ,\:2acos\phi >[/tex]

Using

[tex]\int \int F\cdot dS\:=\:\int \int F\cdot \left(r_u\:x\:r_v\right)\:dA[/tex]

I get

[tex]F\cdot \left(r_u\:x\:r_v\right)\:=a^3sin^3\phi cos^2\theta +a^3sin^3\phi sin^2\theta +2a^3sin\phi cos^2\phi =a^3sin\phi cos^2\phi [/tex]

[tex]\int _0^{2\pi }\int _0^{\frac{\pi }{2}}a^3sin\phi cos^2\phi \:d\phi d\theta [/tex]

[tex]=\frac{2}{3}\pi a^3[/tex]

which is a factor of 4 off, as the answer is [tex]=\frac{8}{3}\pi a^3\gamma [/tex]

*not including the electric constant

Thanks,

RedDelicious

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