Solved Examples of Maxwell's Equations in Differential Form

📚 Quick Study Guide

• ⚡ Gauss's Law for Electricity: Relates the electric field to the electric charge. $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$, where $\mathbf{E}$ is the electric field, $\rho$ is the charge density, and $\epsilon_0$ is the permittivity of free space.
• 🧲 Gauss's Law for Magnetism: States that there are no magnetic monopoles. $\nabla \cdot \mathbf{B} = 0$, where $\mathbf{B}$ is the magnetic field.
• 🔄 Faraday's Law of Induction: Describes how a changing magnetic field creates an electric field. $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$, where $t$ is time.
• 🌊 Ampère-Maxwell's Law: Relates the magnetic field to electric current and changing electric fields. $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, where $\mu_0$ is the permeability of free space, and $\mathbf{J}$ is the current density.
• 💡 Displacement Current: Maxwell's addition to Ampère's Law, $\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, accounts for changing electric fields acting as a current.
• 🧭 Constitutive Relations: Material-dependent relations that connect $\mathbf{D}$ to $\mathbf{E}$ and $\mathbf{H}$ to $\mathbf{B}$. For linear, isotropic materials: $\mathbf{D} = \epsilon \mathbf{E}$ and $\mathbf{B} = \mu \mathbf{H}$.
• ➕ Superposition: The total field due to multiple sources is the vector sum of the fields due to each individual source.

Practice Quiz

1. What does $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$ represent?
   1. Gauss's Law for Magnetism
   2. Ampère's Law
   3. Gauss's Law for Electricity
   4. Faraday's Law
2. Which equation implies the non-existence of magnetic monopoles?
   1. $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$
   2. $\nabla \cdot \mathbf{B} = 0$
   3. $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
   4. $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$
3. Faraday's Law of Induction is represented by which equation?
   1. $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$
   2. $\nabla \cdot \mathbf{B} = 0$
   3. $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
   4. $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$
4. What is the significance of the displacement current term in Ampère-Maxwell's Law?
   1. It accounts for the magnetic force on moving charges.
   2. It accounts for the electric current due to moving charges.
   3. It accounts for the changing electric fields acting as a current.
   4. It accounts for the magnetic field due to permanent magnets.
5. In the context of Maxwell's equations, what do the constitutive relations describe?
   1. The relationship between electric and magnetic fields in vacuum.
   2. The relationship between electric and magnetic fields in materials.
   3. The relationship between charge density and current density.
   4. The relationship between energy density and power density.
6. Which of Maxwell's equations describes how a changing magnetic field induces an electric field?
   1. Gauss's Law for Electricity
   2. Gauss's Law for Magnetism
   3. Faraday's Law of Induction
   4. Ampère-Maxwell's Law
7. What is the differential form of Ampère's Law with Maxwell's addition?
   1. $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$
   2. $\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$
   3. $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$
   4. $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$