Poisson’s and Laplace’s Equations Quiz Quiz

Challenge your understanding of Poisson’s and Laplace’s equations with scenario-based questions covering boundary conditions, solution properties, and physical interpretations common in mathematics, physics, and engineering. This quiz is designed to reinforce essential concepts and distinctions crucial for solving electrostatics and potential field problems.

1. Identifying Laplace’s Equation in Electrostatics

   In a region of space with no free charge, which equation correctly represents Laplace’s equation for an electrostatic potential V(x, y, z)?

   1. ∇·E = 0
   2. ∇²V = ρ/ε₀
   3. ∇²V = 0
   4. ∇V = 0

   Explanation: Laplace’s equation is given by ∇²V = 0, which applies when there is no free charge in the region. The option ∇²V = ρ/ε₀ is the form for Poisson’s equation, where ρ is the charge density. The choice ∇·E = 0 describes the divergence of the electric field, not the equation for potential. ∇V = 0 incorrectly suggests a constant potential in all directions, not Laplace’s condition.

2. Poisson’s Equation in a Capacitor Scenario

   Consider a parallel-plate capacitor filled with a uniform charge density ρ between the plates. Which equation describes the potential V inside the capacitor?

   1. ∇²V = 0
   2. ∇²V = ρ/ε₀
   3. ∇·V = ρ
   4. ∇V = ρ/ε₀

   Explanation: Inside a region with uniform charge density, the potential satisfies Poisson’s equation, which is ∇²V = ρ/ε₀. The distractor ∇V = ρ/ε₀ is dimensionally incorrect as it relates gradient and scalar quantities improperly. ∇·V = ρ is not the correct operation for scalar potential. ∇²V = 0 applies only when the charge density is zero, which is not the case in this scenario.

3. Physical Meaning of Laplace’s Equation

   What does Laplace’s equation imply about the distribution of sources or sinks in the region where it is satisfied?

   1. There are no sources or sinks of the field in that region
   2. There is a uniform distribution of sources
   3. The field strength is always zero
   4. The potential must be constant throughout the region

   Explanation: Laplace’s equation, ∇²V = 0, indicates the absence of sources or sinks (such as charge in electrostatics) in the region. The field can exist but does not require the potential to be constant everywhere, disputing the third option. The field strength itself is not always zero; this is incorrect, as gradients of the potential may exist. A uniform distribution of sources contradicts the statement of Laplace’s equation, where the source term is specifically zero.

4. Boundary Conditions for Unique Solutions

   Which of the following must be specified on the boundary of a region to obtain a unique solution to Laplace’s equation for the potential?

   1. Only the value of the electric field
   2. Both the potential and its second derivative
   3. Only the potential at the center of the region
   4. Either the value of the potential or the normal derivative of the potential

   Explanation: To ensure a unique solution to Laplace’s equation, boundary conditions must specify either the value (Dirichlet) or the normal derivative (Neumann) of the potential on the boundary. Only specifying the electric field is insufficient, as it may not determine the potential uniquely. Both the potential and its second derivative are not generally required together. Knowledge of the potential only at the center does not constrain the solution adequately.

5. Maximum Principle for Laplace’s Equation

   Which statement about solutions to Laplace’s equation in a bounded domain is correct according to the maximum principle?

   1. The maximum and minimum values of the solution occur only on the boundary
   2. The minimum value is always negative
   3. The solution must be zero everywhere inside the domain
   4. The maximum value always occurs at the center of the domain

   Explanation: The maximum principle states that solutions to Laplace’s equation achieve their maximum and minimum values at the boundary, unless the solution is constant within the domain. The other options are incorrect because the solution is not always zero or negative inside. The center does not generally correspond to the extremum unless the boundary conditions dictate it.