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fluidistic
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Homework Statement
A sphere of radius [tex]a[/tex] has a distribution of charge [tex]\rho (r)=Cr[/tex] where [tex]C[/tex] is a constant.
1)Calculate the electric field in all points of space.
2)Calculate the electric potential outside the sphere.
Homework Equations
To figure out.
The Attempt at a Solution
I got all wrong, I'd like to know where are my mistakes.
1) The electric field is radial. To calculate it outside the sphere I first calculate the total charge [tex]Q[/tex] of the sphere : [tex]Q= \int _0^a \rho (r) dr = \int _0 ^a Crdr= \frac{Ca^2}{2}[/tex].
Now I imagine a sphere (Gaussian surface) of radius [tex]r>a[/tex] and I have that [tex]\Phi=\frac{Q_{\text{enclosed}}}{\varepsilon _0} = \oint \vec E d \vec A=EA=E4 \pi r^2[/tex] but we already saw that [tex]Q_{\text{enclosed}}= \frac{Ca^2}{2}[/tex], thus [tex]E4 \pi r^2 = \frac{Ca^2}{2 \varepsilon _0} \Rightarrow E= \frac{Ca^2}{8 \pi \varepsilon _0 r^2}[/tex], [tex]r>a[/tex]. (wrong result)
Now I calculate the electric field inside the sphere : [tex]\Phi=\oint \vec E d \vec A = 4 \pi r^2 E = \frac{Cr^2}{2 \varepsilon _0} \Rightarrow E= \frac{C}{8 \pi \varepsilon _0}[/tex] for [tex]r<a[/tex]. (wrong result, as you can see I got that the electric field inside the sphere does not depend on [tex]r[/tex]... impossible!)
2)[tex]E=- \nabla \varphi[/tex].
Thus [tex]\varphi (r)=- \int _a ^r E(r)dr = - \int _a ^r \frac{Ca^2}{8 \pi \varepsilon _0 r^2}dr=\frac{Ca^2}{8 \pi \varepsilon _0} \left [ \frac{1}{r}- \frac{1}{a} \right ][/tex], with [tex]r>a[/tex] where [tex]r[/tex] is the distance between the center of the sphere and the considered point outside the sphere.
Which is also wrong.