Separation of Variables to Calculate Potential Inside Box

In summary, the conversation discusses a homework problem involving calculating electric potentials inside a cubical box with multiple plates at different potentials. The method of separation of variables is being used and the group is having trouble with the bottom plate having potential as well. Their teacher has mentioned a method called "Fourier's trick" to help with this difficulty.
  • #1
Seraphim755
1
0
My friends and I are in our first senior-level physics course at the University of Alabama in Huntsville, Introductory E&M. At the moment, we're working on using separation of variables to calculate electric potentials inside different objects given certain boundary conditions. One, however, is giving us problems.

1. Homework Statement


A cubical box of side length a consists of four metal plates that are welded together and grounded. The top and bottom faces of the cube are made of separate metal sheets and insulated from the others. These faces are held at a constant potential V0. Find the potential inside the box.

Homework Equations



Standard form for separation of variables for this problem leads to the form:
X(x) = Asin(kx) + Bcos(kx)
Y(y) = Csin(ly) + Dcos(ly)
Z(z) = Ee√(k2+l2)z + Fe-√(k2+l2)z

With variables being changed based on the boundary conditions in the problems, which can be found by considering which plates are grounded. For this particular problem, we believe the boundary conditions to yield:

V = 0 @ x = 0 , x = a
V = 0 @ y = 0 , y = a
V = V0 @ z = 0 , z = a

The Attempt at a Solution


[/B]
We worked a similar problem where only one plate had potential, the top plate. It was fairly straightforward, with the X and Y portions turning into sine functions and the Z portion becoming a hyperbolic sine function thanks to the exponentials. However, we are not sure what to do with this given the bottom plate having potential as well. Our teacher mentioned a method that he referred to as "Fourier's trick" that is a bit much to try to type out. Not sure if this is well-known terminology, but I thought I'd mention it.

Thanks in advance for the help, we deeply appreciate it!
 
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  • #2
Seraphim755 said:
My friends and I are in our first senior-level physics course at the University of Alabama in Huntsville, Introductory E&M. At the moment, we're working on using separation of variables to calculate electric potentials inside different objects given certain boundary conditions. One, however, is giving us problems.

1. Homework Statement


A cubical box of side length a consists of four metal plates that are welded together and grounded. The top and bottom faces of the cube are made of separate metal sheets and insulated from the others. These faces are held at a constant potential V0. Find the potential inside the box.

Homework Equations



Standard form for separation of variables for this problem leads to the form:
X(x) = Asin(kx) + Bcos(kx)
Y(y) = Csin(ly) + Dcos(ly)
Z(z) = Ee√(k2+l2)z + Fe-√(k2+l2)z

With variables being changed based on the boundary conditions in the problems, which can be found by considering which plates are grounded. For this particular problem, we believe the boundary conditions to yield:

V = 0 @ x = 0 , x = a
V = 0 @ y = 0 , y = a
V = V0 @ z = 0 , z = a

The Attempt at a Solution


[/B]
We worked a similar problem where only one plate had potential, the top plate. It was fairly straightforward, with the X and Y portions turning into sine functions and the Z portion becoming a hyperbolic sine function thanks to the exponentials. However, we are not sure what to do with this given the bottom plate having potential as well. Our teacher mentioned a method that he referred to as "Fourier's trick" that is a bit much to try to type out. Not sure if this is well-known terminology, but I thought I'd mention it.

Thanks in advance for the help, we deeply appreciate it!
Consider the symmetry of Z with respect to z, assuming z = 0 is at the center of the box.

Chet
 
  • #3
What exactly is the difficulty you're running into?
 

1. What is separation of variables?

Separation of variables is a mathematical technique used to solve differential equations by breaking down a complex problem into simpler sub-problems. It involves assuming that the solution can be expressed as a product of functions of different variables and then solving each part separately.

2. How is separation of variables used to calculate potential inside a box?

In the context of electromagnetic theory, separation of variables is used to solve the Laplace equation, which describes the electric potential inside a box. By assuming the solution can be expressed as a product of functions of the three spatial coordinates, the equation can be simplified and solved for each individual function.

3. What are the benefits of using separation of variables?

Separation of variables allows for the solution to be broken down into simpler sub-problems, making it easier to solve. It also provides a systematic approach to solving differential equations and can be applied to a wide range of problems in physics and engineering.

4. Are there any limitations to using separation of variables?

Yes, separation of variables is only applicable to certain types of differential equations, specifically those that are linear and homogeneous. It also requires the boundary conditions of the problem to be homogeneous, meaning they are equal to zero.

5. Can separation of variables be used for problems in other fields of science?

Yes, separation of variables is a widely used technique in various fields of science, including physics, engineering, and mathematics. It can be applied to problems involving diffusion, heat transfer, fluid dynamics, and more.

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