Transcription of Solutions to Laplace’s Equations
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Solutions to Laplace’s Equations
1 We have seen that in the region of space where there are no sources of charge, laplace s equation gives the potential. In this lecture, we will discuss Solutions of laplace s equation subject to some boundary conditions. Formal Solution in One Dimension The solution of laplace s equation in one dimension gives a linear potential, has the solution , where m and c are constants. The solution is featureless because it is a monotonically increasing or a decreasing function of x. Further, since the potential at x is the average of the potentials at x+a and x-a, it has no minimum or maximum. This feature, as we will subsequently see, is common to the Solutions in higher dimensions as well. Formal Solution in Two Dimensions In two dimensions , the laplace s equation is Solutions to laplace s Equations Lecture 14: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, , Bombay 2 Since the derivatives are partial, we must have where a is a constant.
In the next lecture we will obtain solutions of Laplace’s equation in spherical coordinates. 7 . Tutorial Assignment . 1. Find an expression for the potential in the region between two infinite parallel planes , the potential on the planes being given by the following : 2. Obtain a solution to Laplace’s equation in two dimensions in ...
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