Methods | Part IB, 2004

Write down the general form of the solution in polar coordinates (r,θ)(r, \theta) to Laplace's equation in two dimensions.

Solve Laplace's equation for ϕ(r,θ)\phi(r, \theta) in 0<r<10<r<1 and in 1<r<1<r<\infty, subject to the conditions

ϕ0 as r0 and rϕr=1+=ϕr=1 and ϕrr=1+ϕrr=1=cos2θ+cos4θ.\begin{gathered} \phi \rightarrow 0 \quad \text { as } \quad r \rightarrow 0 \text { and } r \rightarrow \infty \\ \left.\phi\right|_{r=1+}=\left.\phi\right|_{r=1-} \quad \text { and }\left.\quad \frac{\partial \phi}{\partial r}\right|_{r=1+}-\left.\frac{\partial \phi}{\partial r}\right|_{r=1-}=\cos 2 \theta+\cos 4 \theta . \end{gathered}

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