Laplace's Equation To Find Potential

[baltimorebest]

Homework Statement

Consider a box that has a top and bottom at a/2 and –a/2, while the sides are located at –b/2 and +b/2. Also, the top and bottom are at potential V₀=100V and the sides have V=0. You will need to use the separation of variables technique.

1) Find the general solution using standard methods.
2) Use the given boundary conditions to solve for the solution for this particular geometry.
3) Graph the potential as a function of x/b for y/a=1/2 as a function of y/a for
x/b=0. Also do it for x/b at y/a=0,1/4 and for y/a at x/b=1/2, 1.

Homework Equations

The Attempt at a Solution

I need help with the majority of this problem. My professor gave us the hint of using separation of variables, and she also gave us an example in the book to follow. However, I am still confused on how to do it. As far as the third part of this question, I don't really understand the question. Please help me, and thanks ahead of time.

 

[diazona]

This is very similar (but not quite identical) to the problem being discussed at https://www.physicsforums.com/showthread.php?t=436541, so you might want to follow along there as well.

 

[baltimorebest]

Oh ok. I will follow along and try to post when the other person does. Thanks.

 

[baltimorebest]

I tried to work on this all afternoon and here is what I got for everything except the graphing. I have no idea if it's correct, but I really hope it's close.

 

[paranormal]

Laplace's equation is a fundamental equation in electromagnetism and is used to find the potential in a given region. In this problem, we are given a box with specific boundary conditions and are asked to find the potential using separation of variables.

To begin, we can write Laplace's equation as ∇²V = 0, where V is the potential and ∇² is the Laplacian operator. Using the separation of variables technique, we can assume that the potential can be written as V(x,y) = X(x)Y(y), where X and Y are functions of x and y respectively.

Substituting this into Laplace's equation and rearranging, we get X''(x)/X(x) + Y''(y)/Y(y) = 0. Since the left side of the equation only depends on x and the right side only depends on y, both sides must be equal to a constant, which we will call -λ. Therefore, we have two separate equations: X''(x) + λX(x) = 0 and Y''(y) - λY(y) = 0.

Solving these equations, we get X(x) = A cos(√λx) + B sin(√λx) and Y(y) = Ce^√λy + De^-√λy. Combining these solutions and using the given boundary conditions, we can find the general solution for V(x,y).

To solve for the particular solution for this geometry, we can use the given boundary conditions at the top and bottom of the box. Since V(x,a/2) = V(x,-a/2) = 100V, we can set Y(a/2) = Y(-a/2) = 0, which gives us the condition √λ = nπ/a, where n is a positive integer. Substituting this into our general solution, we get V(x,y) = ∑ A_n cos(nπx/a) sinh(nπy/a), where A_n is a constant.

To graph the potential as a function of x/b for y/a=1/2, we can simply substitute y=a/2 into our solution and plot it as a function of x/b. Similarly, for x/b=0 we can substitute x=0 into our solution and plot it as a function of y/a. For the other cases, we