Question

The function \( f(x, y) \) satisfies the Laplace equation
\[ \nabla^2 f(x, y) = 0 \] on a circular domain of radius \( r = 1 \) with its center at point P with coordinates \( x = 0, y = 0 \). The value of this function on the circular boundary of this domain is equal to 3.
The numerical value of \( f(0, 0) \) is:

• A. 0
• B. 2
• C. 3
• D. 1

Hint:

When solving the Laplace equation on a circular domain, use symmetry and apply the boundary conditions to find the solution. The value at the center is often equal to the average of the boundary values in the case of steady-state solutions.

Answer 1

The Correct Option is C

We are given that the function \( f(x, y) \) satisfies the Laplace equation on a circular domain. The general form of the Laplace equation is: \[ \nabla^2 f(x, y) = 0 \] This is a partial differential equation, and it describes the behavior of the function \( f(x, y) \) within the circular domain. The Laplace equation is a special case of the Poisson equation, and it is commonly used to describe steady-state heat distribution, electrostatic potential, or fluid flow in physics.
The problem provides the following details: - The domain is a circle with radius \( r = 1 \), centered at \( (x, y) = (0, 0) \). - The function \( f(x, y) \) on the boundary of this circle (the boundary is at \( r = 1 \)) has a constant value of 3. Step 1: General Solution to the Laplace Equation
The general solution to the Laplace equation in polar coordinates, due to the circular symmetry of the problem, will be of the form: \[ f(r, \theta) = A + Br \] where: - \( A \) and \( B \) are constants determined by the boundary conditions, - \( r \) is the radial distance from the origin, - \( \theta \) is the angular coordinate. Since the problem involves a circular domain, the angular part does not influence the function due to the symmetry, so we can focus only on the radial part.
Step 2: Applying the Boundary Condition
The boundary condition tells us that at \( r = 1 \), the value of the function is equal to 3: \[ f(1, \theta) = 3 \] Substitute this into the general solution: \[ A + B(1) = 3 \] Thus, we get the equation: \[ A + B = 3 \] Step 3: Finding the Value at the Center
To find the value of \( f(0, 0) \), we substitute \( r = 0 \) into the general solution: \[ f(0, 0) = A + B(0) = A \] From the boundary condition, we have \( A + B = 3 \), and since the function is constant on the boundary, we can conclude that \( A = 3 \). Therefore: \[ f(0, 0) = 3 \] Thus, the numerical value of \( f(0, 0) \) is 3, making (C) the correct answer.