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Introductory Physics Homework Help
How Does Electron Shape Affect Electric Field Energy?
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[QUOTE="mintsnapple, post: 4866449, member: 492363"] [h2]Homework Statement [/h2] a. Calculate the energy density of the electric field at a distance r from an electron (presumed to be a particle) at rest. b. Assume now that the electron is not a point but a sphere of radius R over whose surface the electron charge is uniformly distributed. Determine the energy associated with the external electric field in vacuum of the electron as a function of R. [h2]Homework Equations[/h2] $$ u_e = 1/2\epsilon_0E^2 $$ [h2]The Attempt at a Solution[/h2] a. The electric field of an electron can be assumed to be the same as a point charge, that is $$ E = \frac{q}{4\pi\epsilon_0r^2} $$ Since $$E^2 = \frac{q^2}{16\pi(\epsilon_0)^2r^4} $$, $$u_e = \frac{q^2}{32\pi \epsilon_0 r^4} $$ b. We use Gauss's law to find the electric field of this sphere. $$ EA = \frac{\sigma A}{\epsilon_0} $$ So that $$ E = \frac{\sigma}{\epsilon_0}$$, where $$\sigma$$ is the charge per unit area. So the energy density is $$ u_e = \frac{1}{2}\epsilon_0\frac{\sigma}{(\epsilon_0)^2} = \frac{1}{2} \frac {\sigma^2}{\epsilon_0} $$ The total energy is therefore the energy density multiplied by the volume, so $$ U = \frac{4\sigma^2\pi R^3}{6\epsilon_0} $$ Is this correct? [/QUOTE]
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How Does Electron Shape Affect Electric Field Energy?
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