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Published byMay Jefferson Modified over 8 years ago
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Wave Equations: EM Waves
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Electromagnetic waves for E field for B field
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In general, electromagnetic waves Where represents E or B or their components
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# A plane wave satisfies wave equation in Cartesian coordinates # A spherical wave satisfies wave equation in spherical polar coordinates # A cylindrical wave satisfies wave equation in cylindrical coordinates
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Solution of 3D wave equation In Cartesian coordinates Separation of variables
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Substituting for we obtain Variables are separated out Each variable-term independent And must be a constant
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So we may write where we use
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Solutions are then Total Solution is plane wave
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Traveling 3D plane wave
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spherical coordinates
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spherical waves
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Alternatively The wave equation becomes
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Put Then Hence
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Therefore Wave equation transforms to
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Which follows that Separation of variables Solutions are Total solution is
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outgoingwaves incomingwaves Final form of solution General solution spherical wave
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Cylindrical waves
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with angular and azimuthal symmetry, the Laplacian simplifies and the wave equation
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The solutions are Bessel functions. r For large r, they are approximated as
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A plane wave satisfies one-dimensional wave equation in Cartesian coordinates The position vector must remain perpendicular to the given plane
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The wave then satisfies the generalization of the one-dimensional wave equation
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Plane EM waves in vacuum
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Wave vector k is perpendicular to E Wave vector k is perpendicular to B
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B is perpendicular to E
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B, k and E make a right handed Cartesian co-ordinate system
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Plane EM waves in vacuum
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