E_M_Chap3 - R V ~ ∇ ~ A d 3 r = H S ~ A d ~ S...

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A property of solutions of Laplace’s equation This chapter mostly deals with solutions of Laplace’s equation. Elec- trostatic potential obeys Laplace’s equation at point with no charge density. There is a very special property of solutions of Laplace’s equation that I would like to point out. If φ ( ~ r ) obeys Laplace’s equation in a region and if the value of this function φ ( ~ r ) | surface is zero on the bounding surface, then φ ( ~ r ) = 0 for every point inside that region. This is a very powerful theorem and can be proved by the part c) of the problem 1-60. Proof: In divergence theorem, i.e.,
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Unformatted text preview: R V ‡ ~ ∇ · ~ A · d 3 r = H S ~ A · d ~ S , substitute ~ A = φ ~ ∇ φ . This then leads to Z V ‡ ~ ∇ · ‡ φ ~ ∇ φ ·· d 3 r = I S ‡ φ ~ ∇ φ · · d ~ S Z V ( φ ∇ 2 φ + k∇ φ k 2 ) d 3 r = I S ± φ ∂φ ∂n ¶ dS But, since ∇ 2 φ = 0, we get Z V k∇ φ k 2 d 3 r = 0 . However, k∇ φ k 2 is semi-positive definite quantity and integration over the region can be zero only if ∇ φ = 0. Thus φ is a constant function and then from continuity it must be zero at all points inside....
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