These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

### Exercise 1: A Two-Dimensional Case
As a "warm-up", consider the following two-dimensional situation. A squared 'c'-shaped "slot" is set up, where the two parallel horizontal pieces extend from $x=0$ to $x\to \infty$, and the vertical connecting piece sits at $x=0$ and extends from $y=0$ to $y=a$. Both horizontal pieces are grounded, and the vertical piece is held to a potential $V(x=0,y) = V_0(y)$, where $V_0(y)$ is a function to be specified.
![](images/SepVarCartesian/RectangularC.png)
First, set up the solution of Laplace's equation with the specified boundary conditions. Generally, it will be an *infinite series*.
Your solution, which must be quite generic until we specify $V_0(y)$, should include an integral in terms of $V_0(y)$. For simple forms of $V_0(y)$, the integral can be done by hand. But for "interesting" forms of $V_0(y)$, it is valuable to evaluate this integral with numerical techniques.
It will be impossible to evaluate *every* term in an infinite series - you must choose a sufficient number of terms to keep. In each example below, experiment to see how many terms are necessary to capture the solution. You might try $N=5, N=10, N=20\ldots$.
For each of the $V_0(y)$ functions specified below, (1) evaluate your solutions for $V(x,y)$ numerically; (2) produce a *contour plot* of $V(x,y)$; and (3) describe your solution in physical terms - for example, does the behavior of the potential match your expectations?
- $V_0(y) = 6.0\sin\left(\frac{3\pi y}{a}\right)$. **Note**: This example can be easily evaluated by hand, as a way of checking your numerical method.
- $V_0(y) = -y^2 + ay$. **Note**: This example *can* also be evaluated by hand...
- $V_0(y) = \sinh(y - \frac{a}{2})$.
### Exercise 2: A Three-Dimensional Case
For a three-dimensional case, the same overall scheme allows us to solve Laplace's equation in Cartesian coordinates.
Consider a semi-infinite "pipe": at $x=0$ there is a rectangular plate held at a potential (to be specified later) $V(x=0, y,z) = V_0(y,z)$. Four infinitely long plates are joined to the four edges of the first plate, each extending from $x=0$ to $x\to\infty$. The four infinitely long plates are grounded.
![](images/SepVarCartesian/RectangularPipe.png)
Dimensions of the semi-infinite "pipe":
- From $x=0$ to $x\to\infty$.
- From $y=0$ to $y=a$.
- From $z=0$ to $z=b$.
Again, set up the solution of Laplace's equation with the specified boundary conditions. Generally, it will be an *infinite series*. Similarly to the two-dimensional case, you will encounter integrals in terms of $V_0(y,z)$, but now they are double integrals over **both** $y$ and $z$!
For each of the $V_0(y,z)$ functions specified below, (1) evaluate your solutions for $V(x,y,z)$ numerically; (2) produce a *contour plot* of $V(x,y,z)$, in different cross sections for constant $z$; and (3) describe your solution in physical terms - for example, does the behavior of the potential match your expectations?
- $V_0(y,z) = y$.
- $V_0(y,z)=-4y^2 + 4ay - z^2 + bz + \frac{3}{4}ab\, yz$.
- $V_0(y,z)=\sinh\left(\left(y-\frac{a}{2}\right)\,\left(z-\frac{b}{2}\right)\right)$.