Finding Electric Field of Exponential Charge Distribution

In summary, the conversation discusses finding the electric field for a charge distribution and determining the appropriate surface to integrate over. There is confusion about the charge distribution and whether it exists only along the x-axis or in all dimensions. The participants also consider different surfaces, such as a sphere and a cylinder, and question whether Gauss's Law must be used.
  • #1
rodriguez1gv
9
0

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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  • #2
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]?
 
  • #3
[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:
  • #4
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.
 
  • #5
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.
 
  • #6
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?
 

What is an exponential charge distribution?

An exponential charge distribution is a pattern in which the electric charge is distributed in the shape of an exponential curve. This means that the charge density decreases exponentially as you move away from the center.

How do you find the electric field of an exponential charge distribution?

To find the electric field of an exponential charge distribution, you can use the formula E = kλ/r, where k is the Coulomb's constant, λ is the charge density, and r is the distance from the center of the distribution. However, for more complex distributions, you may need to use calculus and the superposition principle to calculate the electric field.

What are some real-life examples of exponential charge distributions?

One example is the electric field surrounding a charged conducting sphere, which follows an exponential distribution. Another example is the electric field inside a capacitor with an exponential distribution of charge on its plates.

How does the electric field change as you move away from the center of an exponential charge distribution?

The electric field decreases as you move away from the center of an exponential charge distribution. This is because the charge density decreases as you move further away, resulting in a weaker electric field.

Why is it important to understand the electric field of an exponential charge distribution?

Understanding the electric field of an exponential charge distribution is important in many fields, including electromagnetism and electronics. It allows us to calculate the force on a test charge, which is crucial in understanding the behavior of charged particles and designing electrical devices.

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