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Separation of Variables in Cartesian Coordinates

Developed by Jordan McDonnell - Published July 16, 2016

This set of exercises will guide the student through solving Laplace's equation for the electric potential in Cartesian coordinates via separation of variables. They will perform numerical integration and produce plots of the electric potential for situations with non-trivial boundary conditions.
Subject Areas Electricity & Magnetism and Mathematical/Numerical Methods Beyond the First Year IPython/Jupyter Notebook Students who complete this set of exercises will - be able to use separation of variables to solve Laplace's equation in Cartesian coordinates in two dimensions (**Exercise 1**), - be able to use separation of variables to solve Laplace's equation in Cartesian coordinates in three dimensions (**Exercise 2**), - be able to supplement an analytical solution with numerical methods, such as numerical integration (**Exercises 1 and 2**), - and be able to produce and analyze visualizations for the electric potential (**Exercises 1 and 2**). 90 min

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? - $V_0(y,z) = y$. - $V_0(y,z)=-4y^2 + 4ay - z^2 + bz + \frac{3}{4}ab\, yz$. - $V_0(y,z)=\sinh\left(\left(y-\frac{a}{2}\right)\,\left(z-\frac{b}{2}\right)\right)$.