Laplace's Equation To Find Potential

In summary, the problem involves a box with potential V0=100V on the top and bottom and a potential of V=0 on the sides. The solution requires using the separation of variables technique. The first step is to find the general solution using standard methods. Then, the given boundary conditions can be used to solve for the solution for this specific geometry. Finally, to graph the potential as a function of x/b for y/a=1/2 and x/b=0,1/4, as well as y/a as a function of x/b for x/b=1/2,1. Some confusion remains on how to approach this problem, but it is being discussed in detail on a forum.
  • #1
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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.
 
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  • #3
Oh ok. I will follow along and try to post when the other person does. Thanks.
 
  • #4
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.
 

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  • #5


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
 

Related to Laplace's Equation To Find Potential

1. What is Laplace's equation for finding potential?

Laplace's equation is a partial differential equation used to model the behavior of scalar fields, such as electric potential or temperature, in a given region of space. It states that the second derivatives of the field with respect to each coordinate add up to zero.

2. How is Laplace's equation used to find potential?

Laplace's equation can be solved using various mathematical methods, such as separation of variables or Fourier series, to find the potential at any point in the given region of space. The solution to Laplace's equation is also known as the potential function.

3. What are the applications of Laplace's equation in science?

Laplace's equation has many applications in science, particularly in fields such as electromagnetism, fluid mechanics, and heat transfer. It is used to model the behavior of electric and gravitational fields, fluid flow, and temperature distributions, among others.

4. What are the boundary conditions for solving Laplace's equation?

The boundary conditions for solving Laplace's equation are the values of the potential function at the boundaries of the given region of space. These values can be specified as either Dirichlet boundary conditions, where the potential is given at the boundary, or Neumann boundary conditions, where the gradient of the potential is given at the boundary.

5. Can Laplace's equation be solved analytically?

Yes, Laplace's equation can be solved analytically using various mathematical techniques. However, in more complex systems, numerical methods may be necessary to find an approximate solution. Additionally, in some cases, an exact analytical solution may not exist and approximations must be made.

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