If the boundary-conditions for the annular plate in Figure 14.1.7 are u(a, u) u0, u(b

Chapter 14, Problem 12

(choose chapter or problem)

If the boundary-conditions for the annular plate in Figure 14.1.7 are

\(u(a, \theta)=u_{0}\),  \(u(b, \theta)=u_{1}\),   \(0<\theta<2 \pi\)

where \(u_{0}\) and \(u_{1}\) are constants, show that the steady-state temperature is given by

\(u(r, \theta)=\frac{u_{0} \ln (r / b)-u_{1} \ln (r / a)}{\ln (a / b)}\)

[Hint: Try a solution of the form \(u(r, \theta)=v(r, \theta)+\psi(r)\).]