Surface Charge Distrib on Plane

[jesuslovesu]

Homework Statement

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

Find the potential.

Homework Equations

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?

 

[R.Harmon]

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.

 

[Moetasim]

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.