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Separation of Variables in Cartesian Coordinates
Developed by J. D. McDonnell
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.
Exercises
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 → ∞, 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) = 𝑉0 (𝑦), where 𝑉0 (𝑦) is a
function to be specified.
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 𝑉0 (𝑦), should include an integral in
terms of 𝑉0 (𝑦). For simple forms of 𝑉0 (𝑦), the integral can be done by hand. But for
"interesting" forms of 𝑉0 (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 ….
For each of the 𝑉0 (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?
3𝜋𝑦
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𝑉0 (𝑦) = 6.0sin (
•
•
checking your numerical method.
𝑉0 (𝑦) = −𝑦 2 + 𝑎𝑦. Note: This example can also be evaluated by hand...
𝑎
𝑉0 (𝑦) = sinh(𝑦 − 2).
𝑎
). Note: This example can be easily evaluated by hand, as a way of
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) = 𝑉0 (𝑦, 𝑧). Four infinitely long plates are joined to the four edges
of the first plate, each extending from 𝑥 = 0 to 𝑥 → ∞. The four infinitely long plates are
grounded.
Dimensions of the semi-infinite "pipe":
•
•
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From 𝑥 = 0 to 𝑥 → ∞.
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 𝑉0 (𝑦, 𝑧), but now they are double integrals over both 𝑦 and 𝑧!
For each of the 𝑉0 (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?
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𝑉0 (𝑦, 𝑧) = 𝑦.
•
𝑉0 (𝑦, 𝑧) = −4𝑦 2 + 4𝑎𝑦 − 𝑧 2 + 𝑏𝑧 + 4 𝑎𝑏 𝑦𝑧.
3
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𝑎
𝑏
𝑉0 (𝑦, 𝑧) = sinh ((𝑦 − 2) (𝑧 − 2)).