Let n be the normal to an equipotential surface at a point P The principal radii of curvature of the surface at P are R 1 and R 2 A formula due to George Green relates normal derivatives ( n n ) of the potential (r) (which satisfies Laplaces equation) at the equipotential surface to the mean curvature of that equipotential surface Derive Greens equation by direct manipulation of Laplaces equation K 1 (R R)

Question: Let n be the normal to an equipotential surface at a point P. The principal radii of curvature of the surface at P are R

Let ˆn be the normal to an equipotential surface at a point P. The principal radii of curvature of the surface at P are R₁ and R₂. A formula due to George Green relates normal derivatives (∂/∂n ≡ ˆn · ∇) of the potential φ(r) (which satisfies Laplace’s equation) at the equipotential surface to the mean curvature of that equipotential surface K = 1 (R + R):

Derive Green’s equation by direct manipulation of Laplace’s equation.

K = 1 (R + R):

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