# Surface Charge Distrib on Plane

• jesuslovesu
In summary, the problem involves finding the potential for a plane charged with a density of \sigma = \sigma_0 sin(\alpha x) sin(\beta y). Gauss's Law cannot be directly applied in this case. Another approach is to use \vec{E}=-\vec{\nabla}.V, where \vec{\nabla}.V=\frac{dV}{dx}\hat{x}+\frac{dV}{dy}\hat{y}+\frac{dV}{dz}\hat{z}. By knowing that the electric field acts in the z direction, the potential can be found by integrating \frac{dV}{dz}=\frac{-\sigma}{2\epsilon_0} with
jesuslovesu

## Homework Statement

The plane z = 0 is charged to a density $$\sigma = \sigma_0 sin(\alpha x) sin(\beta y)$$

Find the potential.

## The Attempt at a Solution

Well I first thing I would normally do is use Gauss's Law to find E
$$E = \frac{\sigma}{2e0}$$ for an infinite plane, however in this case I don't appear to be able to just plug it in like that.

My next thought would be to find the total charge Q, but how does one do that when the plane is infinite?

I think you might have to use $$\vec{E}=-\vec{\nabla}.V$$, where $$\vec{\nabla}.V=\frac{dV}{dx}\hat{x}+\frac{dV}{dy}\hat{y}+\frac{dV}{dz}\hat{z}$$.

So if you know the electric field acts in the z direction you can say that $$\frac{dV}{dz}=\frac{-\sigma}{2\epsilon_0}$$, so you can integrate w.r.t to z to find the potential. I think that's how you would do it anyway, hope this helps.

As a scientist, my approach to solving this problem would involve breaking it down into smaller, more manageable parts. First, I would consider the fact that the plane is charged to a specific density, which is a function of the x and y coordinates. This suggests that the charge is not uniformly distributed across the plane, but rather it varies depending on the location.

Next, I would use the given equation \sigma = \sigma_0 sin(\alpha x) sin(\beta y) to determine the charge density at a specific point on the plane. This can be done by plugging in values for x and y, or by graphing the equation and visually determining the charge density at a given point.

Once I have the charge density at a specific point, I can use Gauss's Law to determine the electric field at that point. This can be done by considering a small Gaussian surface around the point of interest and integrating the electric field over that surface.

To find the potential, I would then use the equation V = -\int_{a}^{b} \vec{E} \cdot d\vec{l}, where a and b represent the start and end points of the path being integrated over. This would allow me to find the potential difference between two points on the plane, and I could repeat this process for multiple points to create a potential map of the plane.

Finally, to find the total charge on the infinite plane, I would use the equation Q = \int_{S} \sigma dS, where S represents the surface of the plane. This integral would take into account the varying charge density across the plane and give me the total charge.

Overall, as a scientist, my approach would involve breaking down the problem into smaller, more manageable parts and using mathematical equations and principles to determine the potential and total charge on the plane.

## 1. What is surface charge distribution on a plane?

Surface charge distribution on a plane refers to the spatial distribution of electric charge over a flat surface. It is a concept used in electrostatics to describe the behavior of electric charges on a conductive or insulating plane.

## 2. How is surface charge distribution on a plane calculated?

Surface charge distribution on a plane can be calculated using the Gauss's law, which states that the electric flux through a closed surface is equal to the net charge enclosed by that surface. The distribution can also be determined using Coulomb's law, which describes the force between two point charges.

## 3. What factors affect surface charge distribution on a plane?

The surface charge distribution on a plane can be affected by various factors, including the material of the plane, the shape of the plane, the distance between charges, and the presence of any external electric fields. The charges on the plane can also redistribute themselves due to the influence of neighboring charges.

## 4. What is the significance of surface charge distribution on a plane in practical applications?

Surface charge distribution on a plane is a crucial concept in many practical applications, such as designing electronic circuits, understanding the behavior of charged particles on conductive surfaces, and predicting the force between two parallel plates. It also plays a role in the study of capacitors and the behavior of dielectric materials.

## 5. How does surface charge distribution on a plane affect the electric field around it?

The surface charge distribution on a plane determines the electric field around it. The electric field lines are perpendicular to the surface of the plane and are stronger where the charge density is higher. The distribution of charges on the plane also affects the direction and magnitude of the electric field, which can be used to manipulate the behavior of charged particles or objects in its vicinity.

• Introductory Physics Homework Help
Replies
4
Views
3K
• Introductory Physics Homework Help
Replies
26
Views
545
• Introductory Physics Homework Help
Replies
2
Views
4K
• Introductory Physics Homework Help
Replies
5
Views
751
• Introductory Physics Homework Help
Replies
11
Views
363
• Introductory Physics Homework Help
Replies
11
Views
665
• Introductory Physics Homework Help
Replies
9
Views
1K
• Introductory Physics Homework Help
Replies
2
Views
885
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
485