Contradiction in Gauss's Law: Analytical vs Intuitive Evaluation

In summary, the contradiction is that the flux through the top of the cylinder is greater than the flux through the bottom.
  • #1
r16
42
0
I was thinking about gauss's law and ran into this contradiction.

Consider this situation in electrostatics. You have an infinite line of charge, uniform charge density [text]\lambda[/tex]. In cgs units, E at a distance r from the line of charge along the +y axis (assuming a left handed coordinate system) is [tex]\frac{2 \lambda}{r}[/tex].

Now consider a right circular cylinder in empty space as a gaussian shape. Intuitively, I will tell you that the net flux is 0 through the entire shape: all field lines that flow into the shape flow out.

But thinking about this analytically gives you a different answer:

Flux through top bottom (closest to the charge) = [tex]-\frac{2 \lambda}{r} (\pi r^2)[/tex]
Flux through sides = 0
Flux through top (furthest away from the charge) = [tex]\frac{2 \lambda}{r + l} (\pi r^2)[/tex]

so the net flux is tex]-\frac{2 \lambda}{r} (\pi r^2) + \frac{2 \lambda}{r + l} (\pi r^2)[/tex] which must be less than 0 because [tex] r > r+l [/tex], which would mean that there is a net negative flux, or some sort of negative charge enclosed in the gaussian surface which is contradictory to the intuitive evaluation of the shape.

Any resolution to this contradiction?
 
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  • #2
It sounds like you have an infinite line of charge, say along the x-axis. Then you want to consider a cylindrical gaussian surface in empty space above the line charge, say along the y-axis.

r16 said:
I was thinking about gauss's law and ran into this contradiction.

Consider this situation in electrostatics. You have an infinite line of charge, uniform charge density [tex]\lambda[/tex]. In cgs units, E at a distance r from the line of charge along the +y axis (assuming a left handed coordinate system) is [tex]\frac{2 \lambda}{r}[/tex].
OK.
Now consider a right circular cylinder in empty space as a gaussian shape. Intuitively, I will tell you that the net flux is 0 through the entire shape: all field lines that flow into the shape flow out.
That's true.

But thinking about this analytically gives you a different answer:

Flux through top bottom (closest to the charge) = [tex]-\frac{2 \lambda}{r} (\pi r^2)[/tex]
Flux through sides = 0
Flux through top (furthest away from the charge) = [tex]\frac{2 \lambda}{r + l} (\pi r^2)[/tex]
Realize that the field from the line charge is radially outward. Thus the field is not perpendicular to the top and bottom--you must consider the angle to calculate the flux through the ends. Similarly, the field is not parallel to the sides--again you must consider the angle to calculate the flux through the sides.

Done correctly, the net flux will be zero.
 
  • #3
rgr that,

opening up my book and looking at it again, I see what you are saying.

I guess I am having difficulty imaging an electric field and calculating values for an electric field especially when it comes to asymmetric shapes, such as an electric field in an unsymmetrical hollowed out sphere or the electric field inside a hollow cubic pipe for example. I could do it using the definition of an electric field
[tex] dE = \int \frac{\rho(\vec{x_n}) d^3 x}{r^2} \hat{r} [/tex]
but this leads to some unnecessarily complicated integrals. I'm sure if I picked a few clever Gaussian surfaces I could solve the problems a whole lot easier, but I have a hard time picking out those surfaces because I have a hard time intuitively picturing what E will look like so I make poor decisions for my surface I'm using
 
  • #4
Gauss's Law technique (i.e. constructing a gaussian surface) is useful in highly symmetric situation. This is where you can construct a surface where the electric field flux is either a constant, or zero, or a combination of the two. A uniform infinite line charge is one such example, where a cylindrical gaussian surface will give a zero net flux at the flat surfaces, and a constant flux at the curved surface. This is why we use Gauss's law method here.

If you cannot construct such a surface with the geometry of the charge given, then the method is no longer useful as far as getting an analytic solution to get the field. You have to use other techniques.

Zz.
 

1. What is the contradiction in Gauss's Law?

The contradiction in Gauss's Law refers to the discrepancy between the analytical and intuitive evaluations of the law. The analytical evaluation uses mathematical equations to determine the electric field, while the intuitive evaluation relies on the concept of flux to determine the electric field.

2. How does the analytical evaluation of Gauss's Law work?

The analytical evaluation of Gauss's Law involves using mathematical equations, such as the divergence theorem, to calculate the electric field at a given point. This method is based on the concept of charge density and assumes a continuous distribution of charge.

3. How does the intuitive evaluation of Gauss's Law work?

The intuitive evaluation of Gauss's Law involves using the concept of flux, which is a measure of the flow of a vector field through a surface. This method is based on the idea that the electric field lines emanating from a positive charge must end on a negative charge, resulting in a net flux of zero through a closed surface.

4. What causes the contradiction in Gauss's Law?

The contradiction in Gauss's Law is caused by the different assumptions and methods used in the analytical and intuitive evaluations. While the analytical evaluation assumes a continuous distribution of charge, the intuitive evaluation does not take into account the discrete nature of charge. This can lead to different results for the electric field at a given point.

5. How can the contradiction in Gauss's Law be resolved?

The contradiction in Gauss's Law can be resolved by understanding the limitations and assumptions of both the analytical and intuitive evaluations. It is important to recognize that both methods have their own strengths and weaknesses and can be used in different scenarios. Additionally, incorporating the concept of charge discretization in the analytical evaluation can help reconcile the differences between the two methods.

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