The Electromagnetic Field
Vector fields, gauge invariance, and Maxwell's equations from the Lagrangian
🔗Course Connections
Video Lecture
Electromagnetic Field - MIT QFT Lecture
Classical electromagnetic field theory from the Lagrangian formulation
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4.1 The Electromagnetic Lagrangian
The electromagnetic field is described by a 4-vector potential Aμ(x) = (φ, A), where φ is the scalar potential and A is the vector potential.
💡Why a 4-Vector Potential?
In special relativity, space and time must be treated on equal footing. The electric potential φ and magnetic vector potential A naturally combine into a 4-vector Aμ = (φ, A1, A2, A3).
This ensures the theory is Lorentz invariant - different observers will agree on the physics despite measuring different E and B fields!
The Lagrangian density for the free electromagnetic field is:
where the field strength tensor is defined as:
This is antisymmetric: Fμν = -Fνμ, giving us 6 independent components corresponding to the 3 components of E and 3 components of B.
4.2 The Field Strength Tensor
In matrix form, the field strength tensor is:
The electric and magnetic fields are extracted as:
4.3 Maxwell's Equations from the Lagrangian
The Euler-Lagrange equation for Aμ is:
For our Lagrangian ℒ = -¼FμνFμν, this yields:
These are Maxwell's equations in vacuum (in the absence of sources)!
Maxwell's Equations (source-free):
- • ∂μFμν = 0 → Gauss's law: ∇·E = 0
- • ∂μFμν = 0 → Ampère-Maxwell: ∇×B - ∂tE = 0
- • ∂μF̃μν = 0 → No magnetic monopoles: ∇·B = 0
- • ∂μF̃μν = 0 → Faraday's law: ∇×E + ∂tB = 0
4.4 Gauge Invariance
A crucial property of electromagnetism is gauge invariance. The Lagrangian and physics are unchanged under the transformation:
where α(x) is an arbitrary scalar function. This is because:
💡Physical Meaning of Gauge Invariance
Gauge invariance reflects a deep truth: potentials are not directly observable. Only the field strengths E and B (or Fμν) are physical.
We have freedom to choose different potentials Aμ that give the same E and B. This "gauge freedom" is actually a symmetry of nature, and by Noether's theorem, it leads to charge conservation!
4.5 Gauge Fixing
While gauge invariance is a physical symmetry, it causes problems for quantization (redundant degrees of freedom). We must "fix the gauge" by imposing a constraint:
Common Gauge Choices:
- • Lorenz Gauge: ∂μAμ = 0Manifestly Lorentz covariant, good for relativistic calculations
- • Coulomb Gauge: ∇·A = 0A0 = 0, transverse polarizations only, good for non-relativistic QED
- • Temporal Gauge: A0 = 0Simplifies time evolution but breaks manifest Lorentz invariance
- • Axial Gauge: nμAμ = 0Useful in specific calculations, nμ is a constant 4-vector
4.6 Classical vs. Quantum Electromagnetism
Classical EM vs. Quantum EM (QED)
How quantization changes our understanding of electromagnetic fields
| Aspect | Classical EM | Quantum EM (QED) |
|---|---|---|
| Fields | E(x,t) and B(x,t) - classical vectors | — |
| Sources | ρ(x,t) and j(x,t) - charge/current densities | — |
| Photons | Emerge as wave packets | — |
| Gauge Invariance | A^μ → A^μ + ∂^μα | — |
| Equations | Maxwell equations | — |
4.7 Coupling to Matter
To include charged particles (e.g., electrons), we add a coupling term to the Lagrangian:
where jμ = (ρ, j) is the 4-current density. For a Dirac fermion ψ:
This gives the complete QED Lagrangian:
⚠️Common Mistakes to Avoid
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📚 Supplementary Video Lectures
For a comprehensive treatment of classical electrodynamics that complements this field theory approach, see the Advanced Physics: Classical Electrodynamics playlist (62 lectures covering Maxwell's equations, wave propagation, boundary conditions, radiation, and transmission lines).
View All 62 Advanced Physics Videos →
Maxwell's Equations & Fundamentals (7 lectures)
Video Lecture
Introduction and Equation of Continuity
Foundation of electromagnetic theory
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Video Lecture
Maxwell's Postulate: Displacement Current
The missing piece in Ampère's law
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Video Lecture
Physical Interpretation of Maxwell's Postulate
Understanding displacement current
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Video Lecture
Illustrations of Displacement Current
Examples and applications
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Video Lecture
Maxwell's Field Equations - Part 1
Integral and differential forms
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Video Lecture
Maxwell's Field Equations - Part 2
Derivations and symmetries
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Video Lecture
Maxwell's Field Equations - Part 3
Complete formulation
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Energy & Momentum in EM Fields (8 lectures)
Video Lecture
Poynting Theorem - Part 1
Energy conservation in EM fields
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Video Lecture
Poynting Theorem - Part 2
Applications and examples
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Video Lecture
Poynting Vector - Part 1
Energy flux in EM waves
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Video Lecture
Poynting Vector - Part 2
Calculations and interpretations
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Video Lecture
Poynting Vector - Part 3
Advanced applications
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Video Lecture
Maxwell Stress Tensor - Part 1
Momentum in EM fields
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Video Lecture
Maxwell Stress Tensor - Part 2
Force and pressure
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Video Lecture
Maxwell Stress Tensor: Radiation Pressure
Pressure from EM waves
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Wave Propagation (10 lectures)
Video Lecture
Plane EM Waves in Free Space - Part 1
Wave equation solutions
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Video Lecture
Plane EM Waves in Free Space - Part 2
Polarization and properties
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Video Lecture
Waves in Isotropic Dielectric - Part 1
Material effects on propagation
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Video Lecture
Waves in Isotropic Dielectric - Part 2
Refractive index and dispersion
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Video Lecture
Waves in Anisotropic Dielectric - Part 1
Birefringence and crystals
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Video Lecture
Waves in Anisotropic Dielectric - Part 2
Double refraction
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Video Lecture
Waves in Conducting Medium - Part 1
Skin depth and attenuation
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Video Lecture
Waves in Conducting Medium - Part 2
Complex refractive index
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Video Lecture
Waves in Conducting Medium - Part 3
Applications to metals
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Video Lecture
Interaction of EM Waves with Matter
Absorption, scattering, dispersion
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Boundary Conditions & Optical Phenomena (7 lectures)
Video Lecture
Boundary Conditions for D & B
Discontinuities at interfaces
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Video Lecture
Boundary Conditions for E & H
Tangential and normal components
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Video Lecture
Laws of Reflection and Refraction
Snell's law from Maxwell equations
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Video Lecture
Fresnel's Formulae - Part 1
Amplitude coefficients
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Video Lecture
Fresnel's Formulae - Part 2
Polarization effects
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Video Lecture
Reflection and Transmission Coefficients
Energy conservation at boundaries
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Video Lecture
Brewster's Law and Polarization
Polarization by reflection
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Video Lecture
Total Internal Reflection Polarization
Evanescent waves
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Scattering of EM Waves (3 lectures)
Video Lecture
Differential Scattering Cross-section
Quantifying scattering processes
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Video Lecture
Thomson Scattering by Free Charge
Classical electron scattering
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Video Lecture
Thomson Scattering: Unpolarized Waves
Cross section calculations
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Electrodynamics of Moving Charges (16 lectures)
Video Lecture
Electrodynamic Potentials - Part 1
Potentials for time-varying sources
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Video Lecture
Electrodynamic Potentials - Part 2
Gauge transformations
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Video Lecture
Retarded Potentials - Part 1
Causality and finite propagation speed
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Video Lecture
Retarded Potentials - Part 2
Green's functions
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Video Lecture
Liénard-Wiechert Potentials - Part 1
Potentials of moving point charge
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Video Lecture
Liénard-Wiechert Potentials - Part 2
Derivation and properties
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Video Lecture
Fields in Uniform Motion - Part 1
Lorentz contracted fields
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Video Lecture
Electric Field in Uniform Motion
Transformation of E field
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Video Lecture
Magnetic Field in Uniform Motion
Transformation of B field
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Video Lecture
Fields in Uniform Motion - Part 4
Complete field structure
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Video Lecture
Fields in Arbitrary Motion - Part 1
Acceleration effects
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Video Lecture
Fields in Arbitrary Motion - Part 2
Velocity and acceleration fields
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Video Lecture
Fields in Arbitrary Motion - Part 3
Radiation zones
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Video Lecture
Radiation from Accelerated Charge - Part 1
Larmor formula
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Video Lecture
Radiation from Accelerated Charge - Part 2
Angular distribution
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Dipole Radiation (6 lectures)
Video Lecture
Oscillating Electric Dipole - Introduction
Antenna basics
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Video Lecture
Dipole: Vector and Scalar Potentials
Potential calculations
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Video Lecture
Dipole: Magnetic Field
Near and far field B
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Video Lecture
Dipole: Electric Field
Near and far field E
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Video Lecture
Dipole: Power Radiation
Radiated power and pattern
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Special Topics (5 lectures)
Video Lecture
Lorentz Force and Potentials
Force in terms of A^μ
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Video Lecture
Potentials in Uniform Fields
Gauge choices for static fields
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Video Lecture
Transmission Line - Introduction
Distributed circuits
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Video Lecture
Transmission Line Equations
Telegrapher's equations
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Video Lecture
Reflection Coefficient
Impedance matching
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Video Lecture
Transmission Line (continued)
Applications
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Video Lecture
Standing Waves and SWR
Voltage standing wave ratio
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Full playlist: Advanced Physics - Classical Electrodynamics (62 lectures covering comprehensive EM theory)
🎯 Key Takeaways
- The EM field is described by a 4-vector potential Aμ = (φ, A)
- Physical fields E and B are encoded in the antisymmetric tensor Fμν
- Maxwell's equations emerge from the Euler-Lagrange equations
- Gauge invariance Aμ → Aμ + ∂μα is a fundamental symmetry
- Gauge fixing is required for quantization (Lorenz, Coulomb, etc.)
- Coupling to matter introduces the interaction term -jμAμ