Electromagnetic Equations
1. Maxwell’s Equations (Foundation of Electromagnetism)
Maxwell’s equations describe the behavior of electric fields (\(E\)), magnetic fields (\(B\)), electric charge (\(\rho\)), and current (\(J\)).
| Equation | Name | Differential Form | Integral Form | Meaning |
|---|---|---|---|---|
| Gauss’s Law | Electric Fields | \(\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}\) | \(\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}\) | Electric field flux through a closed surface is proportional to enclosed charge. |
| Gauss’s Law for Magnetism | Magnetic Fields | \(\nabla \cdot \mathbf{B} = 0\) | \(\oint_S \mathbf{B} \cdot d\mathbf{A} = 0\) | Magnetic monopoles don’t exist; net magnetic flux is always zero. |
| Faraday’s Law | Electromagnetic Induction | \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\) | \(\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{A}\) | A changing magnetic field creates an electric field (basis for electric generators). |
| Ampère’s Law (with Maxwell’s correction) | Magnetic Fields & Current | \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) | \(\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} + \mu_0 \varepsilon_0 \frac{d}{dt} \int_S \mathbf{E} \cdot d\mathbf{A}\) | Electric currents and changing electric fields create magnetic fields. |
2. Lorentz Force Equation (Force on a Charged Particle)
The Lorentz Force describes how an electric and magnetic field acts on a moving charge.
\[ \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \]
Example
A proton ($ q = 1.6 ^{-19} C $) moving at $ v = 10^6 $ m/s perpendicular to a magnetic field of $ B = 0.1 $ T experiences a force:
\[ F = q v B = (1.6 \times 10^{-19}) (10^6) (0.1) = 1.6 \times 10^{-14} N \]
3. Wave Equation for Electromagnetic Waves
From Maxwell’s equations, we derive the wave equation:
\[ \frac{\partial^2 \mathbf{E}}{\partial t^2} = c^2 \nabla^2 \mathbf{E}, \quad \frac{\partial^2 \mathbf{B}}{\partial t^2} = c^2 \nabla^2 \mathbf{B} \]
where $ c = $ is the speed of light.
Example: Electromagnetic Waves in Vacuum
- If $ = E_0 (kx - t) $, then $ $ is perpendicular to $ $.
- The wave moves at speed $ c $.
4. Energy in Electromagnetic Fields
Poynting Vector (Energy Flow)
The power per unit area carried by an EM wave:
\[ \mathbf{S} = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B}) \]
Electromagnetic Energy Density
Total energy per unit volume in an EM field:
\[ u = \frac{1}{2} \left( \varepsilon_0 E^2 + \frac{B^2}{\mu_0} \right) \]
Example: Energy Density of an EM Wave
For an EM wave with $ E = 1000 $ V/m and $ B = 3.33 ^{-6} $ T:
\[ u = \frac{1}{2} \left( (8.85 \times 10^{-12}) (1000)^2 + \frac{(3.33 \times 10^{-6})^2}{4\pi \times 10^{-7}} \right) = 4.42 \times 10^{-6} \text{ J/m}^3 \]
5. Electromagnetic Induction (Faraday’s & Lenz’s Laws)
Faraday’s Law of Induction
\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \] where $ _B = B A $ is the magnetic flux.
Lenz’s Law
The induced current flows in a direction that opposes the change in flux.
Example
A coil with 10 turns experiences a changing magnetic field from 0.5 T to 0.1 T in 2 s, and has an area $ A = 0.01 m^2 $.
\[ \mathcal{E} = -N \frac{d\Phi_B}{dt} = -10 \times \frac{(0.1 - 0.5) \times 0.01}{2} = 0.02 V \]
6. Electromagnetic Radiation Pressure
The radiation pressure exerted by an electromagnetic wave is:
\[ P = \frac{S}{c} = \frac{E \times B}{\mu_0 c} \]
For solar radiation with $ S = 1361 W/m^2 $:
\[ P = \frac{1361}{3 \times 10^8} = 4.54 \times 10^{-6} \text{ N/m}^2 \]
7. Summary Table of Electromagnetic Equations
| Equation Name | Formula | Description |
|---|---|---|
| Gauss’s Law | \(\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}\) | Electric flux is proportional to charge. |
| Gauss’s Law for Magnetism | \(\nabla \cdot \mathbf{B} = 0\) | No magnetic monopoles exist. |
| Faraday’s Law | \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\) | A changing magnetic field induces an electric field. |
| Ampère’s Law | \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) | Electric currents and changing electric fields generate magnetic fields. |
| Lorentz Force | \(\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})\) | Force on a charged particle. |
| Wave Equation | \(\frac{\partial^2 \mathbf{E}}{\partial t^2} = c^2 \nabla^2 \mathbf{E}\) | Describes EM wave propagation. |
| Poynting Vector | \(\mathbf{S} = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B})\) | Energy flow of an EM wave. |
| Induced EMF | \(\mathcal{E} = -\frac{d\Phi_B}{dt}\) | Voltage induced by changing magnetic flux. |