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EMLecture5

# EMLecture5 - Lecture 5 Notes 95.657 Electromagnetic Theory...

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Lecture 5 Notes, 95.657 Electromagnetic Theory I, Fall 2011 Dr. Christopher S. Baird, UMass Lowell 1. The Laplace Equation in Spherical Coordinates - In this coordinate system, r is the radial distance from the origin to the observation point, θ is the polar angle that the point makes with the z -axis, and ϕ is the azimuthal angle in the x - y plane relative to the x -axis. - Spherical coordinates are useful when the boundary conditions have a spherical shape or symmetry. - The Laplace equation in spherical coordinates: 2 = 0 1 r 2 r 2 r  1 r 2 sin ∂  sin ∂ ∂ 1 r 2 sin 2 2 ∂ 2 = 0 - Use the method of separation of variables by trying a solution of the form:  r , , = R r r P  Q  Here an extra factor (1/ r ) is included to anticipate that the mathematics will be simplified if each factor has the same dimensionality. - Substitute this into the Laplace equation: 1 r 2 r 2 r [ R r r P  Q  ]  1 r 2 sin ∂  sin ∂ [ R r r P  Q  ] 1 r 2 sin 2 2 ∂ 2 [ R r r P  Q  ] = 0 P  Q  1 r 2 R r r 2 R r r Q  1 r 2 sin ∂  sin P  ∂  R r r P  1 r 2 sin 2 2 Q  ∂  2 = 0 - This equation is complex enough that we can not make each term independent all at once. First, get Q in a form to show it is independent by multiplying by r 3 sin 2 / R r P  Q  : r 2 sin 2 R r d 2 R r d r 2 sin P  d d sin d P  d 1 Q  d 2 Q  d 2 = 0 - Here the partial derivatives have become total derivatives because the functions they operate on are now functions of only one variable. - The last term is now independent of ρ and θ , and must hold for all ρ and θ , so that it must equal a constant: r 2 sin 2 R r d 2 R r d r 2 sin P  d d sin d P  d m 2 = 0 and m 2 = 1 Q  d 2 Q  d 2 - We can solve the second equation. First put it in a more intuitive form:

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d 2 Q  d 2 =− m 2 Q  - Now the general solution is clearly: Q (ϕ)= A m e i m ϕ + B m e im ϕ if m ≠ 0 and Q (ϕ)= A m = 0 + B m = 0 ϕ if m = 0. - The constant m is in general not necessarily an integer. If the region of interest includes the full azimuthal sweep of values, then m must be an integer to keep the solution single-valued and the case of m = 0 reduces to Q ( φ ) = A m =0 . From here on, we are dealing with this special case, which is still quite general.
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