Presentation on theme: "Chris Olm and Johnathan Wensman December 3, 2008"— Presentation transcript:
1Chris Olm and Johnathan Wensman December 3, 2008 The Laplace EquationChris Olm and Johnathan WensmanDecember 3, 2008
2Introduction (Part I)We are going to be solving the Laplace equation in the context of electrodynamicsUsing spherical coordinates assuming azimuthal symmetryCould also be solving in Cartesian or cylindrical coordinatesThese would be applicable to systems with corresponding symmetryBegin by using separation of variablesChanges the system of partial differential equations to ordinary differential equationsUse of Legendre polynomials to find the general solution
3Introduction (Part II) We will then demonstrate how to apply boundary conditions to the general solution to attain particular solutionsExplain and demonstrate using “Fourier’s Trick”Analyzing equations to give us a workable solution
4The Laplace Equation Cartesian coordinates Spherical coordinates V is potentialHarmonic!Spherical coordinatesr is the radius is the angle between the z-axis and the vector we’re considering is the angle between the x-axis and our vector
5Azimuthal Symmetry Assuming azimuthal symmetry simplifies the system In this case, decoupling V from Φbecomes
6Potential Function Want our function in terms of r and θ So Where R is dependent on rΘ is dependent on θ
7ODEs! Plugging in we get: These two ODEs that must be equal and opposite:.↓↓
8General Solution for R(r) AssumePlugging in we getSoWe can deduce that the equation is solved when k=l or k=-(l+1)So our general solution for R(r) is
9General Solution for Θ(θ) Legendre polynomialsThe solutions to the Legendre differential equation, where l is an integerOrthogonalMost simply derived using Rodriques’s formula:In our case x=cosθ so
13Example 1 What is the potential on the inside of the sphere? The potential is specified on a hollow sphere of radius RWhat is the potential on the inside of the sphere?
14Applying boundary conditions and intuition We know it must take the formAnd on the surface of the sphere must be V0, also all Bl must be 0, so we getThe question becomes are there any Al which satisfy this equation?
15Yes! But doing so is tricky First we note that the Legendre polynomials are a complete set of orthogonal functionsThis has a couple consequences we can exploit
16Applying this property We can multiply our general solution by Pl’’(cos θ) sin θ and integrate (Fourier’s Trick)
17Solving a particular case We could plug this into our equation givingIn scientific terms this is unnecessarily cumbersome (in layman's terms this is a hard integral we don’t want or need to do)
18The better wayInstead let’s use the half angle formula to rewrite our potential asPlugging THIS into our equation givesNow we can practically read off the values of Al
19Getting the final answer Plugging these into our general solution we get
20Example 2 Very similar to the first example The potential is specified on a hollow sphere of radius RWhat is the potential on the outside of the sphere?
21Proceeding as before Must be of the form All Al must be 0 this time, and again at the surface must be V0, soand
22Fourier’s Trick againBy applying Fourier’s Trick again we can solve for BlAs far as we can solve without a specific potential
23ConclusionBy solving for the general solution we can easily solve for the potential of any system easily described in spherical coordinatesThis is useful as the electric field is the gradient of the potentialThe electric field is an important part of electrostatics
24References1)Griffiths, David. Introduction to Electrodynamics. 3rd ed. Upper Saddle River: Prentice Hall, 1999.2)Blanchard, Paul, Robert Devaney, and Glen Hall. Differential equations. 3rd ed. Belmont: Thomson Higher Education, 2006.3) White, J. L., “Mathematical Methods Special Functions Legendre’s Equation and Legendre Polynomials,”accessed 12/2/2008.4) Weisstein, Eric W. "Laplace's Equation--Spherical Coordinates." From MathWorld--A 5) Wolfram Web Resource. accessed 12/2/2008