# Gauss's Law in 2D: Developing the Continuity Equation

• Pacopag
In summary, the homework statement is to develop the continuity equation in 2D. The closest analogue to Gauss' law is Stokes Theorem.
Pacopag

## Homework Statement

Is there an analog to Gauss's theorem (sorry about the title) in 2D.
The reason is to develop the continuity equation in 2D.

## Homework Equations

In 3D it is.
$$\int_{S}{\bf A}. d{\bf S}=\int_{V}(\nabla . {\bf A})dV$$
(please forgive my latex. A and dS should be vectors).

## The Attempt at a Solution

I suppose we may extend the situation to 3d by allowing a little slice "out of the page" (so that our volume looks like a pancake) and just use the theorem as usual, while imposing that A does not have any component "out of the page", then afterwards take the thickness of the slice to go to zero.

Last edited:
Sure. That's just the divergence theorem. Replace S by boundary(V). In fact there is a 1D version. It's called the fundamental theorem of calculus.

Gauss' Law is also known as the "divergence theorem". You can get $\vec{A}$ and $\d\vec{S}$ by [ itex ]\vec{A}[ /itex ] and [ itex ]d\vec{A}[ /itex ] without the spaces. In general you can see the latex code by clicking on the LaTex.

The closest analogue to Gauss' law in 2 dimensions is Stokes Theorem:
$$\int_C \vec{v}\cdot ds= \int\int_S (\del\cdot d\vec{S}$$
where C is the boundary of the surface S. If S is in the xy-plane, that is Green's Theorem.

All of those are special cases of the generalized Stoke's theorem:
$$\int_M d\omega= \int_{\partial M} \omega$$
where $\omega$ is a differential form on the simply connected manifold M, $d\omega$ is the differential of $\omega$, and $\partial M$ is the boundary of M.

So in 2D it would look like
$$\int_{L}{\bf A}. d{\bf L}=\int_{V}(\nabla . {\bf A})dS$$
Where S is now the area (taking the place of the volume in the 3D version), and L is the contour enclosing the area (taking the place of the surface in the 3D version), and $$\nabla$$ is the 2D gradient operator. Is that right?

Looking good. What I'm getting is that the continuity equation looks like
$$\nabla.(\rho u)+{{\partial \rho}\over{\partial t}}=0$$
where rho is the density and u is the 2D velocity field.
i.e. exactly the same as the 3D continuity equation, but now the $$\nabla$$ is the 2D gradient operator.
Is all this correct?

Yes. The continuity equation looks the same in all dimensions.

Great. Thank you both very much.

## What is Gauss's Law in 2D?

Gauss's Law in 2D is a mathematical equation used to describe the relationship between the electric field and the charge distribution in a two-dimensional space. It is a simplified version of Gauss's Law in three dimensions and is often used in electrostatics problems.

## How is Gauss's Law in 2D related to the continuity equation?

Gauss's Law in 2D is used to derive the continuity equation, which describes the conservation of charge in a given region of space. This means that the amount of charge entering a region must equal the amount of charge leaving that region, which is an important principle in the study of electricity and magnetism.

## What are the key components of Gauss's Law in 2D?

The key components of Gauss's Law in 2D are the electric flux, which is the amount of electric field passing through a given area, and the charge enclosed within that area. The equation also includes the permittivity of free space, which is a constant that relates the electric field to the charge distribution.

## How is Gauss's Law in 2D applied in real-world scenarios?

Gauss's Law in 2D is used in a variety of real-world scenarios, including designing electrical circuits, analyzing the behavior of charged particles in a magnetic field, and understanding the properties of capacitors and other electrical components.

## What are some common misconceptions about Gauss's Law in 2D?

One common misconception about Gauss's Law in 2D is that it only applies to perfect two-dimensional systems. In reality, it can be used to approximate the behavior of three-dimensional systems in certain situations. Another misconception is that it only applies to electric fields that are perpendicular to the surface. In fact, it can be used for any orientation of the electric field and surface.

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