- #1
zSanityz
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Pretty much worked out, but stuck! Gauss' Law problem
"Consider a charge density distribution in space given by [tex]\rho = \rho_0 e^{-r/a}[/tex], where [tex]\rho_0[/tex] and [tex]a[/tex] are constants. Using Gauss' Law, derive an expression for the electric field as a function of radial distance, [tex]r[/tex]. Sketch the [tex]E[/tex] vs. [tex]r[/tex] graph."
[tex]\oint \vec D \cdot d\vec s = \int \rho dv=Q[/tex]
[tex]E_r = {{\int \rho dv}\over{4 \pi \epsilon R^2}}[/tex]
Now all I'm pretty sure I just need to integrate it through, and I'll be able to isolate [tex]\rho[/tex] and substitute it back in the original equation [tex]\rho = \rho_0 e^{-r/a}[/tex] and finally isolate [tex]E[/tex] for an answer.
I really can't figure out how to integrate this though, if anyone could explain / go through it, that would be really helpful!
Thank you
<3's
Homework Statement
"Consider a charge density distribution in space given by [tex]\rho = \rho_0 e^{-r/a}[/tex], where [tex]\rho_0[/tex] and [tex]a[/tex] are constants. Using Gauss' Law, derive an expression for the electric field as a function of radial distance, [tex]r[/tex]. Sketch the [tex]E[/tex] vs. [tex]r[/tex] graph."
Homework Equations
[tex]\oint \vec D \cdot d\vec s = \int \rho dv=Q[/tex]
[tex]E_r = {{\int \rho dv}\over{4 \pi \epsilon R^2}}[/tex]
Now all I'm pretty sure I just need to integrate it through, and I'll be able to isolate [tex]\rho[/tex] and substitute it back in the original equation [tex]\rho = \rho_0 e^{-r/a}[/tex] and finally isolate [tex]E[/tex] for an answer.
I really can't figure out how to integrate this though, if anyone could explain / go through it, that would be really helpful!
Thank you
<3's
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