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

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Introduction (Part I) We are going to be solving the Laplace equation in the context of electrodynamics We are going to be solving the Laplace equation in the context of electrodynamics Using spherical coordinates assuming azimuthal symmetry 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 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 Use of Legendre polynomials to find the general solution

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Introduction (Part II) We will then demonstrate how to apply boundary conditions to the general solution to attain particular solutions 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

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The Laplace Equation Cartesian coordinates Cartesian coordinates –V is potential –Harmonic! Spherical coordinates 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

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Azimuthal Symmetry Assuming azimuthal symmetry simplifies the system Assuming azimuthal symmetry simplifies the system In this case, decoupling V from Φ In this case, decoupling V from Φbecomes

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

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

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General Solution for R(r) Assume 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

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General Solution for Θ(θ) Legendre polynomials 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 In our case x=cosθ so

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

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The General Solution for V Putting together R(r), Θ(θ) and summing over all l Putting together R(r), Θ(θ) and summing over all lbecomes

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Applying the general solution

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Example 1 The potential is specified on a hollow sphere of radius R The potential is specified on a hollow sphere of radius R What is the potential on the inside of the sphere?

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Applying boundary conditions and intuition We know it must take the form We know it must take the form And on the surface of the sphere must be V 0, also all B l must be 0, so we get And on the surface of the sphere must be V 0, also all B l must be 0, so we get The question becomes are there any A l which satisfy this equation? The question becomes are there any A l which satisfy this equation?

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

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Applying this property We can multiply our general solution by P l ’’ (cos θ) sin θ and integrate (Fourier ’ s Trick) We can multiply our general solution by P l ’’ (cos θ) sin θ and integrate (Fourier ’ s Trick)

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

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

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

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

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Proceeding as before Must be of the form Must be of the form All A l must be 0 this time, and again at the surface must be V 0, so All A l must be 0 this time, and again at the surface must be V 0, so and and

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Fourier ’ s Trick again By applying Fourier ’ s Trick again we can solve for B l By applying Fourier ’ s Trick again we can solve for B l As far as we can solve without a specific potential As far as we can solve without a specific potential

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

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References 1)Griffiths, David. Introduction to Electrodynamics. 3rd ed. Upper Saddle River: Prentice Hall, 1999. 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. 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,” 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/s8l egd/s8legd.html, accessed 12/2/2008. http://www.tmt.ugal.ro/crios/Support/ANPT/Curs/math/s8/s8l egd/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/LaplacesEquationSphericalCoo rdinates.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/LaplacesEquationSphericalCoo rdinates.html, accessed 12/2/2008Weisstein, Eric W.MathWorld http://mathworld.wolfram.com/LaplacesEquationSphericalCoo rdinates.htmlWeisstein, Eric W.MathWorld http://mathworld.wolfram.com/LaplacesEquationSphericalCoo rdinates.html

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