jackson3-2

# jackson3-2 - 2 JACOB LEWIS BOURJAILY We should note that...

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Unformatted text preview: 2 JACOB LEWIS BOURJAILY We should note that this more-or-less agrees with the expressions for the potential worked out in the text for similar circumstances. In particular, Jackson’s equation (3.36) describes a similar problem, where ϕ 7→ ϕ = ϕ- V/ 2 on the inner sphere and the outer sphere is removed completely. Making this redefinition of the potential on the inner two hemispheres and taking the limit where aλb , (keeping only terms O ( a i b j ) where j ≥ i ), we obtain Jackson’s equation (3.36) as desired. Problem 3.2 Let us consider the potential induced by a uniform, spherical charge distribution of radius R , with density Q/ 4 πR 2 except for a spherical cap centered on θ = 0 defined by the cone θ = α , where there is no charge. a) We are to show that the potential inside the spherical surface can be expressed as ϕ ( ρ,θ ) = Q 8 π² ∞ X ‘ =0 1 2 ‘ + 1 ρ ‘ R ‘ +1 [ P ‘ +1 (cos α )- P ‘- 1 (cos α )] P ‘ (cos θ ) . Quite generally, we can express the potential as a Poisson integral over the charge dis- tribution. Specifically, ϕ ( x ) = 1 4 π² Z ρ ( x ) d 3 x | x- x | ....
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jackson3-2 - 2 JACOB LEWIS BOURJAILY We should note that...

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