Consider the steady-state temperature distribution in a circular disc of radius $C$ centere at the origin, with temperature given as a function, $f ( \theta )$ on the boundary. The mathematical model of this situation is
A) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$
B) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$
C) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$
D) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$
E) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$

Question

Quizplus · Accepted Answer

The correct equation for the steady-state temperature distribution in a circular disc in polar coordinates is given by the Laplace equation in polar coordinates: $\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} = 0$, with the boundary condition $u(c, \theta) = f(\theta)$. This equation correctly accounts for the radial and angular components of the Laplacian in polar coordinates.

Consider the Steady-State Temperature Distribution in a Circular Disc of Radius