# Electromagnetic Field Theory Textbook

Written by: Bo Thidé (Uppsala School of Engineering in Sweden)

This free online, advanced physics textbook is written for advanced undergraduates or graduate students. The text is “written from a classical field theoretical point of view, emphasising fundamental and subtle properties of the EM field and includes a comprehensive appendix on the mathematical methods used.”

This textbook has been adopted by several universities in Asia, Europe and the Americas. It was developed from the author’s lecture notes for fourth-year Master degree students studying engineering physics at the Uppsala School of Engineering in Sweden. A second volume contains exercises.

The author, Bo Thidé, is affiliated with the Swedish Institute of Space Physics, the Department of Physics and Astronomy at Uppsala University and the Galilean School of Higher Education at the University of Padua. He was preparing to release a revised and expanded 2nd edition when I last visited this site. In mid-2015 it was still being described as a draft release with a date of 18 December 2012. However, the text appears to be a complete edition.

1 Foundations of Classical Electrodynamics
1.1 Electrostatics

1.1.1 Coulomb’s law
1.1.2 The electrostatic field
1.2 Magnetostatics
1.2.1 Ampère’s law
1.2.2 The magnetostatic field
1.3 Electrodynamics
1.3.1 The indestructibility of electric charge
1.3.2 Maxwell’s displacement current
1.3.3 Electromotive force
1.3.5 The microscopic Maxwell equations
1.3.6 Dirac’s symmetrised Maxwell equations

1.4 Examples
1.5 Bibliography
2 Electromagnetic Fields and Waves
2.1 Axiomatic classical electrodynamics
2.2 Complex notation and physical observables
2.2.1 Physical observables and averages
2.2.2 Maxwell equations in Majorana representation
2.3 The wave equations for E and B
2.3.1 The time-independent wave equations for E and B
2.4 Examples
2.5 Bibliography
3 Electromagnetic Potentials and Gauges
3.1 The electrostatic scalar potential
3.2 The magnetostatic vector potential

3.3 The electrodynamic potentials
3.4 Gauge conditions
3.4.1 Lorenz-Lorentz gauge
3.4.2 Coulomb gauge
3.4.3 Velocity gauge
3.5 Gauge transformations
3.5.1 Other gauges
3.6 Examples
3.7 Bibliography
4 Fundamental Properties of the Electromagnetic Field
4.1 Discrete symmetries
4.1.1 Charge conjugation, spatial inversion, and time reversal
4.1.2 Csymmetry
4.1.3 Psymmetry
4.1.4 Tsymmetry
4.2 Continuous symmetries
4.2.1 General conservation laws
4.2.2 Conservation of electric charge
4.2.3 Conservation of energy
4.2.4 Conservation of linear (translational) momentum
4.2.4.1 Gauge-invariant operator formalism
4.2.5 Conservation of angular (rotational) momentum
4.2.5.1 Gauge-invariant operator formalism
4.2.6 Electromagnetic duality
4.2.7 Electromagnetic virial theorem
4.3 Examples
4.4 Bibliography
5 Fields from Arbitrary Charge and Current Distributions
5.1 Fourier component method
5.2 The retarded electric field
5.3 The retarded magnetic field
5.4 The total electric and magnetic fields at large distances from the sources
5.4.1 The far fields
5.5 Examples
5.6 Bibliography
6.1 Radiation of linear momentum and energy
6.1.1 Monochromatic signals
6.1.2 Finite bandwidth signals
6.3 Radiation from a localised source at rest
6.3.1 Electric multipole moments
6.3.2 The Hertz potential
6.4 Radiation from an extended source volume at rest
6.4.1 Radiation from a one-dimensional current distribution
6.5 Radiation from a localised charge in arbitrary motion
6.5.1 The Liénard-Wiechert potentials
6.5.2 Radiation from an accelerated point charge
6.5.2.1 The differential operator method
6.5.2.2 The direct method
6.5.2.3 Small velocities
6.5.3 Bremsstrahlung
6.5.4.3 Radiation in the general case
6.5.4.4 Virtual photons
6.6 Examples
6.7 Bibliography
7 Relativistic Electrodynamics
7.1 The special theory of relativity
7.1.1 The Lorentz transformation
7.1.2 Lorentz space
7.1.2.1 Radius four-vector in contravariant and covariant form
7.1.2.2 Scalar product and norm
7.1.2.3 Metrictensor
7.1.2.4 Invariant line element and proper time
7.1.2.5 Four-vector fields
7.1.2.6 The Lorentz transformation matrix
7.1.2.7 The Lorentz group
7.1.3 Minkowski space
7.2 Covariant classical mechanics
7.3 Covariant classical electrodynamics
7.3.1 The four-potential
7.3.2 The Liénard-Wiechert potentials
7.3.3 The electromagnetic field tensor
7.4 Bibliography
8. Electromagnetic Fields and Particles
8.1 Charged particles in an electromagnetic field
8.1.1 Covariant equations of motion
8.1.1.1 Lagrangian formalism
8.1.1.2 Hamiltonian formalism
8.2 Covariant field theory
8.2.1 Lagrange-Hamilton formalism for fields and interactions
8.2.1.1 The electromagnetic field
8.2.1.2 Other fields
8.3 Bibliography
9. Electromagnetic Fields and Matter
9.1 Maxwell’s macroscopic theory
9.1.1 Polarisation and electric displacement
9.1.2 Magnetisation and the magnetising field
9.1.3 Macroscopic Maxwell equations
9.2 Phase velocity, group velocity and dispersion
9.3 Radiation from charges in a material medium
9.4 Electromagnetic waves in a medium
9.4.1 Constitutive relations
9.4.2 Electromagnetic waves in a conducting medium
9.4.2.1 The wave equations for E and B
9.4.2.2 Plane waves
9.4.2.3 Telegrapher’s equation
9.5 Bibliography
￼￼F Formulæ
F.1 Vector and tensor fields in 3D Euclidean space
F.1.1 Cylindrical circular coordinates
F.1.1.1 Base vectors
F.1.1.2 Directed line element
F.1.1.3 Directed area element
F.1.1.4 Volume element
F.1.1.5 Spatial differential operators
F.1.2 Spherical polar coordinates
F.1.2.1 Base vectors
F.1.2.2 Directed line element
F.1.2.3 Solid angle element
F.1.2.4 Directed area element
F.1.2.5 Volume element
F.1.2.6 Spatial differential operators
F.1.3 Vector and tensor field formulæ
F.1.3.1 The three-dimensional unit tensor of rank two
F.1.3.2 The 3D Kronecker delta tensor
F.1.3.3 The fully antisymmetric Levi-Civita tensor
F.1.3.4 Rotational matrices
￼￼F.1.3.5 General vector and tensor algebra identities
F.1.3.6 Special vector and tensor algebra identities
F.1.3.7 General vector and tensor calculus identities
F.1.3.8 Special vector and tensor calculus identities
F.1.3.9 Integral identities
F.2 The electromagnetic field
F.2.1 Microscopic Maxwell-Lorentz equations in Dirac’s symmetrised form
F.2.1.1 Constitutive relations
F.2.2 Fields and potentials
F.2.2.1 Vector and scalar potentials
F.2.2.2 The velocity gauge condition in free space
F.2.2.3 Gauge transformation
F.2.3 Energy and momentum
F.2.3.1 Electromagnetic field energy density in free space
F.2.3.2 Poynting vector in free space
￼￼F.2.3.3 Linear momentum density in free space
F.2.3.4 Linear momentum flux tensor in free space
F.2.3.5 Angular momentum density around x0 in free space
F.2.3.6 Angular momentum flux tensor around x0 in free space
F.2.4.1 The far fields from an extended source distribution
F.2.4.2 The far fields from an electric dipole
F.2.4.3 The far fields from a magnetic dipole
F.2.4.4 The far fields from an electric quadrupole
F.2.4.5 Relationship between the field vectors in a plane wave
F.2.4.6 The fields from a point charge in arbitrary motion
F.3 Special relativity
F.3.1 Metric tensor for flat 4D space
F.3.2 Lorentz transformation of a four-vector
F.3.3 Covariant and contravariant four-vectors
F.3.3.2 Arbitrary four-vector field
F.3.3.3 Four-deloperator
F.3.3.4 Invariant line element
F.3.3.5 Four-velocity
F.3.3.6 Four-momentum
F.3.3.7 Four-current density
F.3.3.8 Four-potential
F.3.4 Fieldtensor
F.4 Bibliography
M Mathematical Methods
M.1 Scalars, vectors and tensors
M.1.1 Vectors
M.1.1.1 Position vector
M.1.2 Fields
M.1.2.1 Scalar fields
M.1.2.2 Vector fields
M.1.2.3 Coordinate transformations
M.1.2.4 Tensor fields
M.2 Vector algebra
M.2.1 Scalar product
M.2.2 Vector product
M.3 Vector calculus
M.3.1 The deloperator
M.3.2 The gradient of a scalar field
M.3.3 The divergence of a vector field
M.3.4 The curl of a vector field
M.3.5 The Laplacian
M.3.6 Vector and tensor integrals
M.3.6.1 First order derivatives
M.3.6.2 Second order derivatives
M.3.7 Helmholtz’s theorem
M.4 Analytical mechanics
M.4.1 Lagrange’s equations
M.4.2 Hamilton’s equations
M.5 Examples
M.6 Bibliography

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Electromagnetic Field Theory Textbook