This question requires identifying the standard mathematical form of the Laplace equation, a fundamental partial differential equation (PDE) in physics and engineering.
Step 1: Understanding the Question:
We need to select the correct formula for the Laplace equation in a two-dimensional Cartesian coordinate system (\(x, y\)).
Step 2: Key Formula or Approach:
The approach is to recall the definition of the Laplacian operator (\(\Delta\) or \(\nabla^2\)) and the structure of the Laplace equation (\(\nabla^2 u = 0\)).
Step 3: Detailed Explanation:
The Laplacian operator in two dimensions is the sum of the second partial derivatives with respect to each spatial variable. For a function \(u(x, y)\), it is:
\[ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \]
The Laplace equation is a homogeneous PDE that states that the Laplacian of a function is zero.
\[ \nabla^2 u = 0 \implies \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]
This equation typically describes steady-state phenomena, such as steady-state heat distribution, electrostatic potentials in charge-free regions, or ideal fluid flow.
Let's analyze the other options:
(C) \( \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \) is the one-dimensional Wave Equation.
(D) \( \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} \) is the one-dimensional Heat Equation (or Diffusion Equation).
Option (B) correctly represents the Laplace equation in two dimensions.
Step 4: Final Answer:
The two-dimensional Laplace equation is \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \).