Hmwk1q1 - 7 Jackson problem 1.9(only the parallel plate part 8 Show using Greens theorem for Dirichlet boundary conditions that the potential

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Homework set 1 1. . Please complete (but do not hand in) the following table: XXXXXXXXX XXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXX Cartesian Cylindrical Spherical deFnitions U = U ( x, y, z ) A = A x e x + A y e y + A z e z A x = A x ( x, y, z ) A y = A y ( x, y, z ) A z = A z ( x, y, z ) gradient U = ( ∂U/∂x ) e x + ( ∂U/∂y ) e y + ( ∂U/∂z ) e z Laplacian 2 U = 2 U ∂x 2 + 2 U ∂y 2 + 2 U ∂z 2 divergence ∇ · A = ∂A x ∂x + ∂A x ∂x + ∂A x ∂x curl ∇ × A = ( ∂A z /∂y - ∂A y /∂z ) e x + ( ∂A x /∂z - ∂A z /∂y ) e y + ( ∂A y /∂x - ∂A x /∂y ) e y Dirac delta δ ( x ) δ ( y ) δ ( z ) 2. Please demonstrate (but do not hand in) the following: 1. ∇ × ∇ U = 0 2. ∇ · ( ∇ × A ) = 0 3. S is a closed surface enclosing V , demonstrate that R V ∇ · Ad 3 x = R S A · n da 4. S is an open surface bounded by the contour C with line element d l , and n is the normal perpendicular to S , demonstrate that R S ( ∇ × A ) · n da = H C A · d l . 1
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3. Jackson (all from second edition) problem 1.1 (parts a and b) 4. Jackson problem 1.3. 5. Jackson problem 1.6 (a, b and c). 6. Jackson problem 1.8
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Unformatted text preview: 7. Jackson problem 1.9 (only the parallel plate part) 8. Show, using Greens theorem for Dirichlet boundary conditions, that the potential within an empty ( = 0) volume of arbitrary shape is if the entire boundary surface is maintained at potential . 9. Jackson problem 1.12. Prove Greens reciprocal theorem : If is the potential due to a volume charge density within a volume V and a surface charge density on the conducting surface S bounding the volume V , while is the potential due to another charge distribution and , then Z V d 3 x + Z S ds = Z V d 3 x + Z S da 10. Jackson problem 1.13 11. Jackson problem 1.14 12. Jackson problem 1.17 2...
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This note was uploaded on 01/08/2012 for the course PHYSICS 707 taught by Professor Electrodynamics during the Fall '11 term at LSU.

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Hmwk1q1 - 7 Jackson problem 1.9(only the parallel plate part 8 Show using Greens theorem for Dirichlet boundary conditions that the potential

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