Finding Electric Field of Exponential Charge Distribution

[rodriguez1gv]

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?

 

[Bryson]

Is $x$ a vector? If not, assume one dimension. Your surface area will most likely be of a sphere. Also, recall that $q_{enc}$ is the total charge. Can you think of another (more formal) way to write $q_{enc}$?

 

[rodriguez1gv]

$x$ 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 $q_{enc}$ can be written as $\int \rho(x)$ I believe. So I should be able to just integrate my charge distribution from $-x$ to $x$ and consider the area a sphere of radius $x$? 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?

 

[Bryson]

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.

 

[rodriguez1gv]

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.

 

[Bryson]

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?