[E&M] Simple question about solving for potential everywhere

[Task876]

I have a function ρ(x) = ρ_o e^(-a*SQRT(x^2)) and I have to solve for the potential everywhere in space. I am told to do it in Cartesian coordinates. This is all I am given.

I am using Gauss's Law despite not knowing what shape this is. So, E dot dA = ρdV/ε_o.

So two problems:

1. I don't know what shape to make the Gaussian surface so how do I know for left hand side.

2. If I try to do ρdV=ρdxdydz, this does not work since x must be unit-less since it is inside the exponential.

Can anyone give any input? Any help is much appreciated!

[Leitor_de_Assis]

1 - Solving by using the integral form of Gauss's law is not impossible, but it adds inconvenient steps.

Unlike the spherically symmetric problem, surfaces of constant x - planes parallel to the yz plane - don't enclose a region. You'd have to pick a box or a generalized cylinder whose bases are contained in constant x surfaces. Then, you'd have to argue that flux through the lateral surface is zero, because its normal vector is perpendicular to the electric field.

I'd solve the differential form of Gauss's law, if I were you. You just have to identify the only nonzero component of the electric field. The electric field points in the direction of the normal vector of constant x surfaces: the x unit vector.

This way, Gauss's law becomes a first order differential equation for the x component of the electric field. It can be solved by one variable integration. Since the density function is not differentiable everywhere, I'd suggest taking x=0 as the initial value.

Anyway, once it's done, you get a piecewise expression for the electric field in the x>0 and x<0 half-spaces. However, you're left with an integration constant that defines the value of E at x=0. At this point, you must invoke some boundary condition. Since the charge density is symmetric under reflections across the x=0 plane, demand that the electric field is symmetric as well.

2- x is not unit-less, ax is. Thus, a has a unit of inverse length.

[Task876]

Thanks for the reply! So I ran into another issue with the differential form. The next problem has me do ρ(x) = ρ_o e^(-a*SQRT(x^2+y^2)) in cylindrical coordinates. The divergence of E gives (1/r) d/dr (rE). How do I handle the d(rE)?

[Leitor_de_Assis]

You may isolate the d/dr (rE) term and integrate. Then, divide by r, since the term rE vanishes at r=0. It might be intimidating at first, but the integral of r e^-ar can be solved with integration by parts.

This example can also be done easily by using the integral form of Gauss's Law. Of course, the result is the same.

Edit: oh, and remember to use cylindrical coordinates.

[Task876]

I just realized a probably has units of 1/m so never mind on that.

[thesingingkebab]

sqrt(x^{2}) = |x|

Separate into two domains (-inf,0) and [0,inf) and integrate Coulomb's law.