- #1
bodensee9
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A sphere of radius R has charge Q distributed uniformly over its surface. How large a sphere contains 90% of the energy stored in the electrostatic field of this charge distribution?
I assume that this is a spherical shell? So, to find energy, would we have
[tex]U = \frac{1}{8\pi}\int E(r)^{2}dV[/tex]
Then, to find E(r), inside the sphere we have no charge since it's a shell?
But outside the sphere we have
[tex]\frac{Q}{r^{2}}\vec{r}[/tex]
Then, if we integrate, would we find that the energy is
[tex]\frac{1}{2}\int\frac{Q^{2}\ast4r^{2}\pi}{r^{4}}dr[/tex]
which is
[tex]\frac{Q^{2}}{2R}[/tex]
So then if we have 90% of the potential energy then we have
[tex]\frac{9Q^{2}}{20R}[/tex]
So would the large sphere have radius 20/9R? Or how do we find such a radius?
Actually, if this is not a spherical shell, but a solid sphere with charge Q distributed uniformly on its surface, would the answer still be the same? Like can we treat the inside as having on charge? Many thanks.
I assume that this is a spherical shell? So, to find energy, would we have
[tex]U = \frac{1}{8\pi}\int E(r)^{2}dV[/tex]
Then, to find E(r), inside the sphere we have no charge since it's a shell?
But outside the sphere we have
[tex]\frac{Q}{r^{2}}\vec{r}[/tex]
Then, if we integrate, would we find that the energy is
[tex]\frac{1}{2}\int\frac{Q^{2}\ast4r^{2}\pi}{r^{4}}dr[/tex]
which is
[tex]\frac{Q^{2}}{2R}[/tex]
So then if we have 90% of the potential energy then we have
[tex]\frac{9Q^{2}}{20R}[/tex]
So would the large sphere have radius 20/9R? Or how do we find such a radius?
Actually, if this is not a spherical shell, but a solid sphere with charge Q distributed uniformly on its surface, would the answer still be the same? Like can we treat the inside as having on charge? Many thanks.