Final Answer:
The steady-state temperature distribution in the given sector with maintained sides is represented by ( T(θ) = T_0 sin²(θ), where ( T_0) is the temperature at the center of the sector.
Step-by-step explanation:
In order to find the steady-state temperature distribution, we can employ the polar coordinate system and Laplace's equation for heat conduction, which is given by (∇ ² T = 0 ). The steady-state temperature distribution ( Tθ) is a function of the polar angle ( θ). The boundary conditions, as mentioned, involve maintaining the sides as shown.
By solving Laplace's equation under these boundary conditions, we arrive at the solution ( Tθ = T_0 sin²θ) ), where ( T_0 ) is the temperature at the center of the sector. This solution satisfies the given boundary conditions and represents the temperature distribution throughout the sector.
The trigonometric function ( sin²(θ) ensures that the temperature is zero at the center ((θ = 0 )) and increases as we move towards the boundaries (θ = pi/2 ). The square term accounts for the fact that heat transfer is proportional to the temperature difference, providing a physically meaningful temperature distribution for the given geometry.
In conclusion, the final expression ( T(θ) = T_0 sin²(θ) ) encapsulates the steady-state temperature distribution in the specified sector, and this solution is derived through the application of Laplace's equation and consideration of the provided boundary conditions.