1. Chris Olm and Johnathan Wensman December 3, 2008
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 Spherical 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. Θ(θ) Part 2
Let l=0: □
Let l=3: ↘ ↘ ↓ ↘ □

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 What is the potential on the inside of the sphere?
The potential is specified on a hollow sphere of radius R
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,”
accessed 12/2/2008.
4) Weisstein, Eric W. "Laplace's Equation--Spherical Coordinates." From MathWorld--A 5) Wolfram Web Resource. accessed 12/2/2008