Methods | Part IB, 2001

Laplace's equation in the plane is given in terms of plane polar coordinates rr and θ\theta in the form

2ϕ1rr(rϕr)+1r22ϕθ2=0\nabla^{2} \phi \equiv \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}=0

In each of the cases

 (i) 0r1, and (ii) 1r<\text { (i) } 0 \leqslant r \leqslant 1, \text { and (ii) } 1 \leqslant r<\infty \text {, }

find the general solution of Laplace's equation which is single-valued and finite.

Solve also Laplace's equation in the annulus arba \leqslant r \leqslant b with the boundary conditions

ϕ=1 on r=a for all θϕ=2 on r=b for all θ.\begin{aligned} &\phi=1 \quad \text { on } \quad r=a \text { for all } \theta \\ &\phi=2 \quad \text { on } \quad r=b \text { for all } \theta . \end{aligned}

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