Separation of Variables to Laplace's Equation in Electrostatics

In summary, the method of separation of variables is utilized to solve Laplace's equation in electrostatics, which describes the behavior of electric potential in a charge-free region. By decomposing the potential into functions of individual spatial variables, one can derive ordinary differential equations that can be solved independently. This approach leads to solutions that satisfy boundary conditions, ultimately providing insight into electric fields and potentials in various geometries, such as plates, cylinders, and spheres. The resulting solutions are essential for understanding electrostatic phenomena in physics and engineering applications.
  • #1
chaos333
11
1
Homework Statement
A cubical box (side length a) consists of five metal plates, which are welded
together and grounded (Fig. 3.23). The top is made of a separate sheet of metal, insulated
from the others, and held at a constant potential V0. Find the potential inside the box. [What
should the potential at the center (a/2, a/2, a/2) be? Check numerically that your formula is
consistent with this value.
Relevant Equations
d^2v/dx^2+d^2v/dy^2+d^2v/dz^2=0
1721178908349.png
1721178923350.png

A bit messy but the bottom is supposed to be the potential function
 
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  • #2
First of all, in the future please use the LaTeX feature rather than posting pictures of writing. Apart from being more legible, it is then possible to quote particular passages of your post.

As for the solution itself: First of all, you are not answering the first question: "What should the potential at the center be?" This is a relatively simple question which is easily answerable.

Second, your constant ##c_n## should be ##c_{nm}## and be inside of the sums.

I did not check the calculations explicitly because they are difficult to read, but in general the approach looks fine. You also need to evaluate at the middle of the box and show numerically that it approaches the correct value.
 

FAQ: Separation of Variables to Laplace's Equation in Electrostatics

What is Laplace's Equation and its significance in electrostatics?

Laplace's Equation is a second-order partial differential equation given by ∇²φ = 0, where φ represents the electric potential. In electrostatics, it is significant because it describes the behavior of electric potentials in regions where there are no free charges. Solutions to Laplace's Equation provide insights into the electric field configurations and potential distributions in electrostatic systems.

How does the method of separation of variables work for solving Laplace's Equation?

The method of separation of variables involves assuming that the solution to Laplace's Equation can be expressed as a product of functions, each depending on a single coordinate. For example, in a two-dimensional case, we might assume φ(x, y) = X(x)Y(y). By substituting this product into Laplace's Equation and rearranging, we can separate the variables, leading to two ordinary differential equations that can be solved independently.

In which coordinate systems can separation of variables be applied to Laplace's Equation?

Separation of variables can be applied in various coordinate systems, including Cartesian, cylindrical, and spherical coordinates. The choice of coordinate system typically depends on the symmetry of the problem at hand. For example, cylindrical coordinates are useful for problems with circular symmetry, while spherical coordinates are suited for problems involving spherical symmetry.

What boundary conditions are typically used when applying separation of variables to Laplace's Equation?

Boundary conditions are crucial for obtaining unique solutions to Laplace's Equation. Common types of boundary conditions include Dirichlet conditions (specifying the potential on the boundary), Neumann conditions (specifying the electric field or potential gradient on the boundary), and mixed conditions. The specific choice of boundary conditions depends on the physical setup of the problem being analyzed.

Can you give an example of a physical problem where separation of variables is used to solve Laplace's Equation?

One classic example is the electrostatic potential in a two-dimensional rectangular region with specified potentials on the boundaries. By applying the method of separation of variables, we can derive the potential distribution within the region. This approach can be used to model scenarios such as the potential around charged conductors or the behavior of electric fields in capacitor configurations.

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