Welcome to the course on Maxwell's Equations. These four fundamental equations govern the behavior of electric and magnetic fields. In this course, we will break down each equation, explain its physical significance, and go through practical calculation examples.
1: Introduction to Maxwell's Equations
Maxwell's Equations are a set of four differential equations that describe how electric fields \(\mathbf{E}\) and magnetic fields \(\mathbf{B}\) interact. They are the foundation of classical electromagnetism, optics, and electric circuits. The equations are:
- Gauss's Law for Electricity – Describes the relation between a static electric field and the distribution of electric charge.
- Gauss's Law for Magnetism – States that there are no "magnetic charges," i.e., the magnetic field lines are always closed loops.
- Faraday's Law of Induction – Describes how a time-varying magnetic field induces an electric field.
- Ampère's Law with Maxwell's Addition – Relates the magnetic field to the electric current and the rate of change of the electric field.
Chapter 2: Gauss's Law for Electricity
Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed within the surface. Mathematically, it is written as:
Equation 1: Gauss's Law for Electricity
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \]
Where:
- \(\mathbf{E}\) = Electric field vector
- \(\rho\) = Charge density
- \(\epsilon_0\) = Permittivity of free space (\(8.854 \times 10^{-12} \, \mathrm{C^2/(N \cdot m^2)}\))
Example Calculation: Electric Flux Through a Sphere
Consider a spherical charge distribution with charge \(Q\) placed at the center of the sphere. The sphere has a radius \(r\), and we want to calculate the electric flux through the surface of the sphere.
Given:
- Charge \(Q = 5 \, \mu C = 5 \times 10^{-6} \, \text{C}\)
- Radius \(r = 0.1 \, \text{m}\)
We use Gauss's law in its integral form:
\[ \Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]
For a spherical surface, the electric field is radial and has the same magnitude at all points on the surface. Thus:
\[ \Phi_E = \frac{Q}{\epsilon_0} \]
Substituting values:
\[ \Phi_E = \frac{5 \times 10^{-6}}{8.854 \times 10^{-12}} \approx 5.65 \times 10^5 \, \text{N} \cdot \text{m}^2/\text{C} \]
Result: The electric flux through the spherical surface is approximately \(5.65 \times 10^5 \, \text{N} \cdot \text{m}^2/\text{C}\).
Chapter 3: Gauss's Law for Magnetism
Gauss's Law for Magnetism states that the magnetic field lines are closed loops, meaning that there are no "magnetic charges" like there are electric charges. This is expressed as:
Equation 2: Gauss's Law for Magnetism
\[ \nabla \cdot \mathbf{B} = 0 \]
This implies that the net magnetic flux through any closed surface is zero, which supports the idea that magnetic monopoles do not exist (at least not as we observe them today).
Chapter 4: Faraday's Law of Induction
Faraday's Law explains how a time-varying magnetic field induces an electric field. It forms the basis for the operation of many devices like transformers and electric generators. The law is expressed as:
Equation 3: Faraday's Law of Induction
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
This equation shows that a changing magnetic field produces a circulating electric field.
Example Calculation: Induced EMF in a Coil
Consider a coil with 100 turns, placed in a changing magnetic field. If the magnetic field changes at a rate of \(\frac{\partial B}{\partial t} = 0.1 \, \text{T/s}\), calculate the induced EMF.
Given:
- Number of turns \(N = 100\)
- Rate of change of magnetic field \(\frac{\partial B}{\partial t} = 0.1 \, \text{T/s}\)
We use the formula for the induced EMF in a coil:
\[ \mathcal{E} = -N \frac{\partial B}{\partial t} \]
Substituting values:
\[ \mathcal{E} = -100 \times 0.1 = -10 \, \text{V} \]
Result: The induced EMF is \(-10 \, \text{V}\).
Chapter 5: Ampère's Law with Maxwell's Addition
Ampère's Law describes how a current and a time-varying electric field generate a magnetic field. Maxwell's addition to Ampère's Law accounts for the changing electric field over time. The law is given by:
Equation 4: Ampère's Law with Maxwell's Addition
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]
Where:
- \(\mathbf{J}\) = Current density
- \(\mu_0\) = Permeability of free space
- \(\epsilon_0\) = Permittivity of free space