Finding Electric Field of Exponential Charge Distribution

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Homework Help Overview

The discussion revolves around finding the electric field for an exponential charge distribution given by the function ρ(x) = e^{−κ√(x²)}. Participants are exploring the implications of the charge distribution's dimensionality and the appropriate application of Gauss's Law.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are questioning whether the charge distribution is one-dimensional or if it extends into the y-z plane. There are discussions about the appropriate surface for integration, with suggestions of using a sphere or a cylinder. Some participants are considering the implications of symmetry in their choice of surface.

Discussion Status

The discussion is active, with participants providing insights and questioning assumptions about the nature of the charge distribution and the application of Gauss's Law. There is no explicit consensus yet, but various approaches are being explored, including comparisons to the electric field of an infinite wire.

Contextual Notes

Participants are grappling with the dimensionality of the charge distribution and the implications for calculating the enclosed charge. There is also uncertainty regarding the most suitable surface for applying Gauss's Law, with considerations of symmetry and charge distribution affecting their reasoning.

rodriguez1gv
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Homework Statement


I am to find the electric field for a charge distribution of
$$ \rho(x)= e^{-\kappa \sqrt{x^2}} $$


Homework Equations



I know that gauss law is $$ \int E \cdot da = \frac{q_{enc}}{\epsilon_0} $$

The Attempt at a Solution



I am not sure what the charge distribution looks like. Is this saying that there is only charge along the x axis? or is the charge everywhere? I am also no sure what kind of surface I should be integrating over. Should I be integrating over a circle and then finding the total charge enclosed within?
 
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Is [itex]x[/itex] a vector? If not, assume one dimension. Your surface area will most likely be of a sphere. Also, recall that [itex]q_{enc}[/itex] is the total charge. Can you think of another (more formal) way to write [itex]q_{enc}[/itex]?
 
[itex]x[/itex] appears to be a scalar. Does this mean that the charge only exists along the x axis? Or is it also distributed through the y-z plane? And the [itex]q_{enc}[/itex] can be written as [itex]\int \rho(x)[/itex] I believe. So I should be able to just integrate my charge distribution from [itex]-x[/itex] to [itex]x[/itex] and consider the area a sphere of radius [itex]x[/itex]? That doesn't seem quite right to me for some reason since I have an x symmetry should I be using a cylinder? similar to a line of charge along the x axis?
 
Last edited:
Yes, really what we have is a point charge in one dimension, where we only consider the charge density along the x-axis. I suppose a cylinder would be fitting for Gauss's Law. Yes, you are correct about integrating along the x-axis.
 
Using a cylinder seems to give me a dependence on both x and y. I feel like there should be a simpler choice of surface, but I cannot seem to think of it. I have also tried a sphere centered at the origin. I am not sure how I would apply a plane.
 
Perhaps, we can treat this similar to the case for an infinite wire? Are we finding the E-field at some point say on the y-axis, or some point on the x-axis?

Must we use Gauss's Law?
 

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