The Laplace Equation

1. The Laplace Equation Chris Olm and Johnathan Wensman December 3, 2008

2. Introduction (Part I) • We are going to be solving the Laplace equation in the context of electrodynamics • Using spherical coordinates assuming azimuthal symmetry • Could also be solving in Cartesian or cylindrical coordinates • These would be applicable to systems with corresponding symmetry • Begin by using separation of variables • Changes the system of partial differential equations to ordinary differential equations • Use of Legendre polynomials to find the general solution

3. Introduction (Part II) • We will then demonstrate how to apply boundary conditions to the general solution to attain particular solutions • Explain and demonstrate using “Fourier’s Trick” • Analyzing equations to give us a workable solution

4. The Laplace Equation • Cartesian coordinates • V is potential • Harmonic! • Spherical coordinates • r 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

5. Azimuthal Symmetry • Assuming azimuthal symmetry simplifies the system • In this case, decoupling V from Φ becomes

6. Potential Function • Want our function in terms of r and θ • So • Where R is dependent on r • Θ is dependent on θ

7. ODEs! • Plugging in we get: • These two ODEs that must be equal and opposite: . ↓ ↓

8. General Solution for R(r) • Assume • Plugging in we get So • We can deduce that the equation is solved when k=l or k=-(l+1) • So our general solution for R(r) is

9. General Solution for Θ(θ) • Legendre polynomials • The solutions to the Legendre differential equation, where l is an integer • Orthogonal • Most simply derived using Rodriques’s formula: • In our case x=cosθ so

10. Let l=0: Let l=3: Θ(θ) Part 2 □ ↘ ↘ ↓ ↘ □

11. The General Solution for V • Putting together R(r), Θ(θ) and summing over all l becomes

12. Applying the general solution

13. Example 1 • The potential is specified on a hollow sphere of radiusR What is the potential on the inside of the sphere?

14. Applying boundary conditions and intuition • We know it must take the form • And on the surface of the sphere must be V0, also all Bl must be 0, so we get • The question becomes are there any Al which satisfy this equation?

15. Yes! But doing so is tricky • First we note that the Legendre polynomials are a complete set of orthogonal functions • This has a couple consequences we can exploit

16. Applying this property • We can multiply our general solution by Pl’’(cos θ) sin θ and integrate (Fourier’s Trick)

17. Solving a particular case • We could plug this into our equation giving • In scientific terms this is unnecessarily cumbersome (in layman's terms this is a hard integral we don’t want or need to do)

18. The better way • Instead let’s use the half angle formula to rewrite our potential as • Plugging THIS into our equation gives • Now we can practically read off the values of Al

19. Getting the final answer • Plugging these into our general solution we get

20. Example 2 • Very similar to the first example • The potential is specified on a hollow sphere of radius R • What is the potential on the outside of the sphere?

21. Proceeding as before • Must be of the form • All Al must be 0 this time, and again at the surface must be V0, so • and

22. Fourier’s Trick again • By applying Fourier’s Trick again we can solve for Bl • As far as we can solve without a specific potential

23. Conclusion • By solving for the general solution we can easily solve for the potential of any system easily described in spherical coordinates • This is useful as the electric field is the gradient of the potential • The electric field is an important part of electrostatics

24. References • 1)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,” • http://www.tmt.ugal.ro/crios/Support/ANPT/Curs/math/s8/s8legd/s8legd.html, accessed 12/2/2008. • 4) Weisstein, Eric W. "Laplace's Equation--Spherical Coordinates." From MathWorld--A 5) Wolfram Web Resource. http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinates.html, accessed 12/2/2008