Classical Electrodynamics

The electric field at the point $ \mathbf{x} $ due to a point charge $ q_1 $ at the point $ \mathbf{x_1} $ can be obtained directly:
$ \mathbf{E(x)} = k \, q_1 \frac{\mathbf{x} - \mathbf{x_1}}{|\mathbf{x} - \mathbf{x_1}|^3} $

Note: The constant k differs in different systems of units.

Pending

Test

Potentials:

$\vec{E} = - \nabla{\phi} - \frac{\partial{\vec{A}} }{\partial t}$
$\vec{B} = \nabla \times \vec{A}$

Equations (2) & (3) are satisfied immediately as identities.

Truly classical models of magnetic susceptibility?

Scalar theory of diffraction?

Scalar and Vector Green functions?

Radiative effects are almost entirely classical in nature?

Maxwell Equations:

$ \nabla . \vec{E} = \frac{1}{\epsilon_0} \rho $
$ \nabla \times \vec{E} = - \frac{\partial{\vec{B}}}{\partial{t}} $
$ \nabla . \vec{B} = 0 $
$ \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial{\vec{E}}}{\partial{t}} $

Mindmap Contents

Guidelines:

Dashed lines connect relevant information
Thiner dashed lines connect section contents
Contents branches refer only to one (the next) level of contents.
Bold: actual commands that Mathematica responds to
Italic: generic terms
[E]: A new label follows with the content of the branch appropriately expanded.
< ... > refers to a variable symbol.
Level 1 Labels are intentionally situated clockwise moving outwards.
In contrast, level 2 & 3 branches often, not always, follow a counter-clockwise direction.

Online Resources:

Sources

Primary Source:

John D. Jackson - Classical Electrodynamics [3rd Edition] (1999) #

Alternative Sources:

David J. Griffiths - Introduction to Electrodynamics [3rd Edition] (1999)
Y. Kahn & A. Anderson - Conquering the Physics GRE [2nd edition] (2017) #

Introduction and Survey #

Introduction and Survey

Introduction

Historical Milestones:

Although amber and lodestone were known to the ancient Greeks, electrodynamics developed as a quantitative subject in less than a hundred years.
Cavendish’s remarkable experiments in electrostatics were done from 1771 to 1773.
Coulomb’s monumental researches began to be published in 1785. This marked the beginning of quantitative research in electricity and magnetism on a worldwide scale.
Fifty years later Faraday was studying the effects of time-varying currents and magnetic fields.
By 1864 Maxwell had published his famous paper on a dynamical theory of the electromagnetic field.
Twenty-four years later (1888) Hertz published his discovery of transverse electromagnetic waves, which propagated at the same speed as light, and placed Maxwell’s theory on a firm experimental footing.

Pending

Questions such as the current accuracy of the inverse square law of force (mass of the photon), the limits of validity of the principle of linear superposition, and the effects of discreteness of charge and of energy differences are discussed. “Bread and butter” topics such as the boundary conditions for macroscopic fields at surfaces between different media and at conductors are also treated. The aim is to set classical electromagnetism in context, to indicate its domain of validity, and to elucidate some of the idealizations that it contains. Some results from later in the book and some non-classical ideas are used in the course of the discussion. Certainly a reader beginning electromagnetism for the first time will not follow all the arguments or see their significance. For others, however, this introduction will serve as a springboard into the later parts of the book, beyond Chapter 5, and will remind them of how the subject stands as an experimental science.

1.1 Maxwell Equations in Vacuum, Fields, and Sources #

Historical Milestones

Maxwell Equations in Vacuum, Fields, and Sources

Maxwell Equations:

The equations governing electromagnetic phenomena are the Maxwell equations:
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where for external sources in vacuum, $ \mathbf{D} = ε_o \mathbf{E} $ and $ \mathbf{B} = μ_o \mathbf{H} $. The first two equations then become:
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Continuity Equation:

Implicit in the Maxwell equations is the continuity equation for charge density and current density:
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This follows from combining the time derivative of the first equation in (1.1a) with the divergence of the second equation.

Proof Outline: $ ε_o μ_o = c^2 $ and $ \nabla . (\nabla \times B) = 0 $

Lorentz Equation:

Also essential for consideration of charged particle motion is the Lorentz force equation:
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which gives the force acting on a point charge $ q $ in the presence of electromagnetic fields.

These equations have been written in SI units.

These definitions assume that the speed of light is a universal constant, consistent with evidence (see Section 11.2.C) indicating that to a high accuracy the speed of light in vacuum is independent of frequency from very low frequencies to at least $ ν = 10^24 $ Hz (4 GeV photons).

The notion of the fields E and B

The electric and magnetic fields $ \mathbf{E} $ and $ \mathbf{B} $ in were originally introduced by means of the force equation. They first appear just as convenient replacements for forces produced by distributions of charge and current, they have other important aspects:

First, their introduction decouples conceptually the sources from the test bodies experiencing electromagnetic forces. If the fields $ \mathbf{E} $ and $ \mathbf{B} $ from two source distributions are the same at a given point in space, the force acting on a test charge or current at that point will be the same, regardless of how different the source distributions are. This gives $ \mathbf{E} $ and $ \mathbf{B} $ in meaning in their own right, independent of the sources.
Second, electromagnetic fields can exist in regions of space where there are no sources. They can carry energy, momentum, and angular momentum and so have an existence totally independent of charges and currents. In fact, though there are recurring attempts to eliminate explicit reference to the fields in favor of action-at-a-distance descriptions of the interaction of charged particles, the concept of the electromagnetic field is one of the most fruitful ideas of physics, both classically and quantum mechanically.

The concept of $ \mathbf{E} $ and $ \mathbf{B} $ as ordinary fields is a classical notion. It can be thought of as the classical limit (limit of large quantum numbers) of a quantum-mechanical description in terms of real or virtual photons.
How is one to decide a priori when a classical description of the electromagnetic fields is adequate? Some sophistication is occasionally needed, but the following is usually a sufficient criterion: When the number of photons involved can be taken as large but the momentum carried by an individual photon is small compared to the momentum of the material system, then the response of the material system can be determined adequately from a classical description of the electromagnetic fields. For example, each $ 10^8 $ Hz photon emitted by our FM antenna gives it an impulse of only $ 2.2 X 10^34 $ N.s. A classical treatment is surely adequate.

There is a lack of symmetry in the appearance of the source terms in the Maxwell equations. The first two equations have sources; the second two do not. This reflects the experimental absence of magnetic charges and currents. Actually, as is shown in Section 6.11, particles could have magnetic as well as electric charge. If all particles in nature had the same ratio of magnetic to electric charge, the fields and sources could be redefined in such a way that the usual Maxwell equations emerge. In this sense it is somewhat a matter of convention to say that no magnetic charges or currents exist. Throughout most of this book it is assumed that only electric charges and currents act in the Maxwell equations, but some consequences of the existence of a particle with a different magnetic to electric charge ratio, for example, a magnetic monopole, are described in Chapter 6.

Through Gauss’s law and the divergence theorem (see Sections 1.3 and 1.4) this leads to the first of the Maxwell equations.

1.2 Inverse Square Law or the Mass of the Photon #

Inverse Square Law or the Mass of the Photon:

Higher precision experiments involving different regimes of size have been performed over the years. It is now customary to quote the tests of the inverse square law in one of two ways:
(a) Assume that the force varies as $ 1/r^{2+ε} $ and quote a value or limit for $ε $.
(b) Assume that the electrostatic potential has the "Yukawa" form $ r^{-1} e^{-μ r} $ and quote a value or limit for $ μ $ or $ μ^{-1} $. Since $ μ = m_γ c/\hbar $, where $ m_γ $ is the assumed mass of the photon, the test of the inverse square law is sometimes phrased in terms of an upper limit on $ m_γ $. Laboratory experiments usually give $ ε $ and perhaps $ μ $ or $ m_γ $; geomagnetic experiments give $ μ $ or $ m_γ $.

A rough limit on the photon mass can be set quite easily by noting the existence of very low frequency modes in the earth-ionosphere resonant cavity (Schumann resonances, discussed in Section 8.9).

Millikan's oil drop eperiment

1.3 Linear Superposition #

Linear Superposition

If the photons are real, the process contributes to the mass of the photon and is decreed to vanish. If the photons are virtual, however, as in the electromagnetic interaction between a nucleus and an orbiting electron, or indeed for any externally applied field, the creation and annihilation of a virtual electron-positron pair from time to time causes observable effects.

The Maxwell equations in vacuum are linear in the fields $ \mathbf{E} $ and $ \mathbf{B} $. This linearity is exploited so often, for example, with hundreds of different telephone conversations on a single microwave link, that it is taken for granted. There are, of course, circumstances where nonlinear effects occur in magnetic materials, in crystals responding to intense laser beams, even in the devices used to put those telephone conversations on and off the microwave beam. But here we are concerned with fields in vacuum or the microscopic fields inside atoms and nuclei.

Evidence for linear superposition:

At the macroscopic level, all sorts of experiments test linear superposition at the level of 0.1% accuracy:

groups of charges and currents produce electric and magnetic forces calculable by linear superposition,
transformers perform as expected,
standing waves are observed on transmission lines.
In optics:
slit systems show diffraction patterns,
x-ray diffraction tells us about crystal structure,
white light is refracted by a prism into the colors of the rainbow and recombined into white light again.
At the macroscopic and even at the atomic level, linear superposition is remarkably valid.

1.4 Maxwell Equations in Macroscopic Media #

D: Electric Displacement
B: Magnetic Induction
H: Magnetic Field
$ ε_o $: Electric Permeability
$ μ_o $: Magnetic Permeability

Maxwell Equations in Macroscopic Media

The macroscopic Maxwell equations are a set of eight equations involving the components of the four ﬁelds $ \mathbf{E} $, $ \mathbf{B} $, $ \mathbf{D} $, and $ \mathbf{H} $. The four homogeneous equations can be solved formally by expressing $ \mathbf{E} $ and $ \mathbf{B} $ in terms of the scalar potential $ Φ $ and the vector potential $ \mathbf{Α} $, but the inhomogeneous equations cannot be solved until the derived ﬁelds $ \mathbf{D} $ and $ \mathbf{H} $ are known in terms of $ \mathbf{E} $ and $ \mathbf{B} $. These connections are known as constitutive relations:$ \mathbf{D} = \mathbf{D[E, B]} $
$ \mathbf{H} = \mathbf{H[E, B]} $
In addition, for conducting media there is the generalized Ohm’s law:

$ \mathbf{J} = \mathbf{J[E, B]} $
The square brackets signify that the connections are not necessarily simple and may depend on past history (hysteresis), may be nonlinear, etc.

In most materials the electric quadrupole and higher terms are completely negligible. Only the electric and magnetic polarizations $ \mathbf{P} $ and $ \mathbf{M} $ are signiﬁcant. This does not mean, however, that the constitutive relations are then simple. There is tremendous diversity in the electric and magnetic properties of matter, especially in crystalline solids, with ferroelectric and ferromagnetic materials having nonzero $ \mathbf{P} $ or $ \mathbf{M} $ in the absence of applied ﬁelds, as well as more ordinary dielectric, diamagnetic, and paramagnetic substances.

In substances other than ferroelectrics or ferromagnets, for weak enough ﬁelds the presence of an applied electric or magnetic ﬁeld induces an electric or magnetic polarization proportional to the magnitude of the applied ﬁeld. We then say that the response of the medium is linear and write the Cartesian components of $ \mathbf{D} $ and $ \mathbf{H} $ in the form:
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The tensors $ \mathbf{ε}_{ab} $ and $ μ^{'}_{αβ} $ are called the electric permittivity or dielectric tensor and the inverse magnetic permeability tensor. They summarize the linear response of the medium and are dependent on the molecular and perhaps crystalline structure of the material, as well as bulk properties like density and temperature. For simple materials the linear response is often isotropic in space. Then $ \mathbf{ε}_{ab} $ and $ μ^{'}_{αβ} $ are diagonal with all three elements equal, and $ \mathbf{D} = \mathbf{ε} \mathbf{E} $, $ \mathbf{H} = \mathbf{μ'} \mathbf{B} = \mathbf{B}/μ$.

The type of response of materials to an applied magnetic field depends on the properties of the individual atoms or molecules and also on their interactions:

Diamagizetic substances consist of atoms or molecules with no net angular momentum. The response to an applied magnetic field is the creation of circulating atomic currents that produce a very small bulk magnetization opposing the applied field. Diamagnetism is a very small effect.
If the basic atomic unit of the material has a net angular momentum from unpaired electrons, the substance is paramagnetic. The magnetic moment of the odd electron is aligned parallel to the applied field.

Ferromagnetic materials are paramagnetic but, because of interactions between atoms, show drastically different behavior. Below the Curie temperature (1040K for Fe, 630K for Ni), ferromagnetic substances show spontaneous magnetization; that is, all the magnetic moments in a microscopically large region called a domain are aligned. The application of an external field tends to cause the domains to change and the moments in different domains to line up together, leading to the saturation of the bulk magnetization. Removal of the field leaves a considerable fraction of the moments still aligned, giving a permanent magnetization that can be as large as $ B_r = μ_O M_r >~ 1 $ tesla.

1.5 Boundary Conditions at Interfaces Between Different Media #

Boundary Conditions at Interfaces Between Different Media

Surface Integrals:

The Maxwell equations are differential equations applying locally at each point in space-time $ (x, t) $. By means of the divergence theorem and Stokes's theorem, they can be cast in integral form. Let $ V $ be a finite volume in space, $ S $ the closed surface (or surfaces) bounding it, $ da $ an element of area on the surface, and $ \mathbf{n} $ a unit normal to the surface at da pointing outward from the enclosed volume. Then the divergence theorem applied to the first and last Maxwell equations yields the integral statements:

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The first relation is just Gauss's law that the total flux of $ \mathbf{D} $ out through the surface is equal to the charge contained inside. The second is the magnetic analog, with no net flux of \mathbf{B} through a closed surface because of the nonexistence of magnetic charges.

Contour Integrals:

Similarly, let $ C $ be a closed contour in space, $ S' $ an open surface spanning the contour, $ dl $ a line element on the contour, $ da $ an element of area on $ S' $, and $ \mathbf{n'} $ a unit normal at $ da $ pointing in the direction given by the right-hand rule from the sense of integration around the contour. Then applying Stokes's theorem to the middle two Maxwell equations gives the integral statements:

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The first equation is the Ampere-Maxwell law of magnetic fields and the second is Faraday's law of electromagnetic induction.

Summary; Electrostatic Boundary Conditions

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Ch. 1: Introduction to Electrostatics #

Introduction to Electrostatics

Chapter 1 Coulomb's Law

All of electrostatics stems from the quantitative statement of Coulomb’s law concerning the force acting between charged bodies at rest with respect to each other.

Ch. 2: Boundary-Value Problems in Electrostatics: I

Ch. 3: Boundary-Value Problems in Electrostatics: II

Ch. 4: Multipoles, Electrostatics of Macroscopic Media, Dielectrics

Ch. 5: Magnetostatics, Faraday's Law, Quasi-Static Fields

Ch. 6: Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws

1. Electrostatics

1.2 Electric Potential

Poisson Equation:

$ \nabla^2 V = -\frac{ρ}{ε_ο} $

$ V(\mathbf{r}) = \frac{1}{4 π ε_ο} \int \frac{ρ(r')}{|\mathbf{r}-\mathbf{r'}|} d^3\mathbf{r'} $

$ \mathbf{E} = - \nabla V $

$ V(b) = - \int_a^b \mathbf{E}.d\mathbf{l} $

Note:

Mind the sign!
This is not potential energy: $ U = q V $

1.3 Integral Form of Maxwell's Equations

Electric flux: $ \oint_S \mathbf{E}(\mathbf{r}) . d \mathbf{S} = \frac{Q_{\text{enclosed}}}{ε_o} $ (From Gauss Theorem)
$ \oint_C \mathbf{E}(\mathbf{r}) . d\mathbf{l} = 0 $ (From Stokes Theorem)

Point Charge:

$ \mathbf{E}(\mathbf{r}) = \frac{q}{4 π ε_ο r^2} \mathbf{\hat{r}} $
$ V(\mathbf{r}) = \frac{q}{4 π ε_ο r} $, (with infinite as the point of reference)

1.4 Standard Electrostatic Configurations

Point Charges:

Dipole potential

Planes:

Infinite flat sheet with surface charge $ σ $. The electric field must point perpendicularly away from the sheet, since this is the only preferred direction:

$ \mathbf{E} = \frac{σ}{2 ε_o} \mathbf{\hat{n}} $

Line Charges and Cylinders:

Infinite line charges -> cylindrical symmetry -> cylindrical Gaussian box.
By symmetry the field can't have a component along the line, and by the Maxwell equation: $ \nabla \times \mathbf{E} = 0 $, the only option is for it to point in the $ \mathbf{\hat{r}} $ direction in cylindrical coordinates:
$ \mathbf{E} = \frac{λ}{2 π ε_o r} \mathbf{\hat{r}} $

Spherical Surfaces:

The electric field inside a spherically symmetric cavity is always zero.

1.5 Boundary Conditions:

$ \mathbf{E^{||}_{out}} -\mathbf{E^{||}_{in}} = 0 $
$ E^{\bot}_{out} - E^{\bot}_{in} = \frac{σ}{ε_o} $

Thus, the electric field is always continuous when there are no surface charges, only volume charges.
V is always continuous
Derivatives of V are continuous, except at charged surfaces.

1.1 Maxwell's Equations for Electrostatics:

$ \nabla . \mathbf{E} = \frac{ρ}{ε_o} $
$ \nabla \times \mathbf{E} = 0 $

1.6 Conductors:

V is constant throughout a conductor

Corollaries:

The electric field inside a conductor is zero.
The net charge density inside a conductor is zero.
Any net charge on a conductor is confined to the surfaces.
The electric field just outside a conductor is perpendicular to the surface.
These properties are a direct consequence of the fact that an ideal conductor has no resistance to the movement of charges.

Idealized Ground in Electrostatics:

Connected to V= 0.
It is n infinite sink and source of charge. So if you put a charge near a grounded conductor, some charge will induced on the grounded conductor with no cost in energy.

1.7 Method of Images

The essence of the method of images is this: whenever we have a conductor, if we can find an arrangement of point charges that exactly reproduces the potential on the surface, then the potential everywhere will be the same as if there were no conductor.

There is one very important subtlety about the method of images. When calculating anything work or energy, remember that there is no energy cost to moving an image charge. The image charge is just fake construct, a trick to get the correct potential; in reality there is not field below the conductor, and so no work can be done on anything below the conductor.

1.8 Work and Energy in Electrostatics

The work required to put together n point charges is:
$ W = \frac{1}{2} \sum^{n}_{i=1} q_i V(\mathbf{r}_i) $
For a continuous distribution of charge:
$ W = \frac{1}{2} \int ρ(\mathbf{r}) V(\mathbf{r}) d^3 \mathbf{r} $

We showed that it takes work to move charge around, but there's also energy stored in the fields themselves. The energy is just:

$ U_E = \frac{ε_o}{2} \int |\mathbf{E}|^2 d^3\mathbf{r} $

1.9 Capacitors:

$ Q = C V $,
C only depends on the geometry of the problem.

Parallel-plate Capacitor:

The total field is $ |\mathbf{E}| = \frac{Q}{A ε_o} $ pointing from the positive to the negative plate.
Doing the line integral along a straight line between the two plates gives: $ V = \frac{Q d}{A ε_o} $.
Therefore $ C = \frac{A ε_o}{d} $.
Energy stored in the field of a charged capacitor is: $ U = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} C V^2 $. This can be interpreted as the energy it takes to remove a charge $ Q $ from one (initially neutral) plate and put it on the other plate.

2. Magnetostatics:

Steady-current approximation.

Note: Electrons move the other way.

2.1 Basic Tools

Maxwell equation in the absence of changing electric fields:
$ \nabla . \mathbf{B} = 0 $
$ \nabla \times \mathbf{B} = μ_o \mathbf{J} $
The first equation is simply the statement that there are no magnetic monopoles, while the second equation is the statemement that currents are sources for magnetic fields.

Integral Form:

Magnetic Flux: $ \oint_S \mathbf{B} . d\mathbf{S} = 0 $
Ampère's Law: $ \oint_C \mathbf{B} . d\mathbf{l} = μ_ο Ι_{encl} $

Vector Potential:

$ \nabla \times \mathbf{A} = \mathbf{B} $

Lorentz Law:

$ \mathbf{F} = q \mathbf{v} \times \mathbf{B} $

Generalizing this force on a wire carrying current I:

$ d\mathbf{F} = I d\mathbf{l} \times \mathbf{B} $

2.2 Ampère's Law and the Biot-Savart Law

Ampère's Law:

Ampère's law is generally only useful for configurations that possess a high degree of symmetry.

Biot-Savart Law:

$ \mathbf{B}(\mathbf{r}) = \frac{μ_o I}{4 π} \int \frac{ d\mathbf{l} \times \hat{r}'}{r'^2} $

2.3 Standard Magnetostatics Configurations

Wires:

Magnetic field at a distance $ r $ from a current-carrying wire:

$ \mathbf{B} = \frac{μ_o I_{encl}}{2 π r} \hat{φ} $

Note: The wire has to be infinite for these symmetry arguments to work.

Solenoids (Coils):

Rectangular Amperian loop with one vertical side of length $ L $ inside the cylinder and the other outside.
Due to symmetry, the field must point along the axis of the cylinder and be constant inside.
If there are $ n $ turns per unit length, then the only nonzero term in Ampère's law is from the one inside the cylinder, which gives you $ B L $.
The current enclosed is $ I n L $, thus: $ B = μ_o n I $.
Applying the same arguments to an Amperian loop outside the cylinder indicates that the field is identically zero outside.
The solenoid confines a strong uniform field to limited volume.

Toroids:

Again due to symmetry the field points along the axis of a cylinder, which is now the $ \hat{φ} $ direction.
Drawing a circular Amperian loop in the plane of the tube, just as in the wire case, give:
$ \mathbf{B} = \frac{μ_ο Ν Ι}{2 π r} \hat{φ} $.
Just as with a solenoid, the field vanishes outside the volume enclosed by the loops of wire.

2.4 Boundary Conditions

2.5 Work and Energy in Magnetostatics

Section 1

1.1 Maxwell's Equations for Electrostatics #

1.2 Electric Potential #

1.3 Integral Form of Maxwell's Equations #

1.4 Standard Electrostatic Configurations #

1.5 Boundary Conditions # #

1.6 Conductors #

1.7 Method of Images #

1.8 Work and Energy in Electrostatics #

1.9 Capacitors #

Sections 2

2.1 Basic Tools #

2.2 Ampère's Law and the Biot-Savart Law #

2.3 Standard Magnetostatics Configurations #

2.4 Boundary Conditions #

2.5 Work and Energy in Magnetostatics #

Temporary Mindmap Contents

Chapter 2: Electricity and Magnetism

Section 2: Magnetostatics #

Section 1: Electrostatics #

Section 3: Electrodynamics #

The normal component of the magnetic field around a surface must be continuous:
$ \boxed{B^{\bot}_{out} - B^{\bot}_{in} = 0} $

The parallel component of the magnetic field around a surface must be continuous:
$ \boxed{\mathbf{E^{||}_{out}} -\mathbf{E^{||}_{in}} = μ_o \mathbf{K} \times \hat{n}} $

A good mnemonic to remember these equations is to notice that they're sort of the reverse of the analgous electrostatic boundaray conditions: normal $ \mathbf{B} $ becomes parallel $ \mathbf{E} $ and vice versa, and $ ε_o $ goes in the denominator while $ μ_ο $ goes in the numerator. Also, note that the surface charge density $ σ $ is scalar, while $ \mathbf{K} $ is a vector; this means that $ σ $ must be related to the scalar $ E^{\bot} $, and $ \mathbf{K} $ is related to the vector $ \mathbf{B}^{||} $.

Magnetic fields do not work:

$ dW = \mathbf{F} d\mathbf{l} = q ( \mathbf{v} \times \mathbf{B} ) d\mathbf{l} = 0 $

2.6 Cyclotron Motion #

2.6 Cyclotron Motion:

Uniform circular motion under a uniform magnetic field. The most general motion in a constant magnetic field is a helix.
Radius of the circle:
$ R = \frac{m u}{q B} $
and the angular frequency (cyclotron frequency):
$ ω = \frac{q B}{m} $.

However, magnetic fields store energy just like the electric fields:
$ U_B = \frac{1}{2 μ_o} \int |\mathbf{B}|^2 d^3 \mathbf{r} $ #

3. Electrodynamics

Section 3

3.1 Maxwell Equations #

3.2 Faraday's Law #

3.2 Faraday's Law

Integrating both sides of $ \nabla \times \mathbf{E} = - \frac{\partial{\mathbf{B}}}{\partial{t}} $ over a surface $ S $ and using Stokes' theorem:

$ \int_S ( \nabla \times \mathbf{E} ) . d\mathbf{S} = \oint_C \mathbf{E} . d\mathbf{l} = - \frac{ \partial}{\partial{t}} ( \int_S \mathbf{B} . d\mathbf{S} ) \equiv - \frac{ \partial}{\partial{t}} Φ_B $

$ Φ_B $ is the magnetic flux through the (not closed) surface $ S $ with boundary $ C $ (a closed curve).

Term $ \oint_C \mathbf{E} . d\mathbf{l} $ is just the electric potential around the loop (up to a sign). Therefore this expression indicates that a changing magnetic flux through a loop of wire sets up a potential (and therefore a current) through the wire, much like a battery would. The electric potential in this context s often called the Electromotive force and is denoted by $ \mathcal{E} $:

$ \boxed{ \mathcal{E} = - \frac{ \partial}{\partial{t}} Φ_B } $

Both $ \nabla \times \mathbf{E} = - \frac{\partial{\mathbf{B}}}{\partial{t}} $ and $ \mathcal{E} = - \frac{ \partial}{\partial{t}} Φ_B $ are referred to as Faraday's law.

3.3 Inductors

3.3 Inductors #

The minus sign in $ \mathcal{E} = - \frac{ \partial}{\partial{t}} Φ_B $ is often referred to as Lenz's law: Induced currents always oppose changes in magnetic flux. Derived from the conservation of energy.

Mutual Inductance:

When two current loops are positioned close to each other, a changing current in one produces a time-varying magnetic field that can influence the other and vice-versa. The flux $ Φ_{21} $ through loop $ 2 $ is proportional to the current $ I_1 $ in loop $ 1 $ via: $ Φ_{12} = M_{12} I_1 $, where $ M_{12} $ is a constant entirely dependent on geometry and known as the mutual inductance. It turns out that $ M_{12} = M_{21} $, so this relationship is symmetric: $ Φ_{12} = M_{21} I_2 = M_{12} I_1 $.

Self-inductance:

The self-inductance (or simply inductance) is generally defined to be the constant $ L $ is generally defined to be the constant $ L $ in the expression: $ Φ_Β = L I $, in which the magnetic flux through an arrangement of wires, where the field is produced by the wires themselves, is proportional to the current carried by the wires.

This immediately implies that: $ \mathcal{E} = - L \frac{dI}{dt} $, where $ I $ is the current.

The self-inductance of an arrangement of wires is the magnetic analogy of the capacitance of an arrangement of conductor, and as with capacitance, the self-inductance is purely determined by the geometry of the arraignment.

Solenoid inductance:

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3.1 Maxwell Equations:

$ \nabla . \mathbf{E} = \frac{\rho}{\epsilon_0} $
$ \nabla . \mathbf{B} = 0 $
$ \nabla \times \mathbf{E} = - \frac{\partial{\mathbf{B}}}{\partial{t}} $ (Inductance)
$ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial{\mathbf{E}}}{\partial{t}} $ (Displacement current)

Section 4: Dipoles #

Dipoles

Section 4

4.1. Electric Dipoles #

4.1. Electric Dipoles

4.2 Magnetic Dipoles

4.2 Magnetic Dipoles #

4.3 Multiple Expansion #

4.3 Multiple Expansion

Definition of Dipole Moment:

Dipole Moment of Configurations and Distributions:

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Potential of a Dipole:

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Motion of the Dipole in a an Electric Field:

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Potential Energy of the Dipole in a an Electric Field:

There are no Magnetic Monopoles:

Definition of the Magnetic Dipole Moment:

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Torque and Potential Energy of a Magentic Dipole in a Magnetic Field:

Note on the Magnetic Field of a Magnetic Dipole:

The Expansion Formula:

$ V(\mathbf{r}) = \frac{1}{4 π ε_ο} \int \frac{ρ(r')}{|\mathbf{r}-\mathbf{r'}|} d^3\mathbf{r'} $
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The Idea Behind Multiple Expansion:

The First Term in the Expansion:

The Second Term in the Expansion:

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A Note on the Multiple Expansion for the Vector Potential:

Matter Effects

Section 5 Section 5: Matter Effects #

5.1 Dielectrics

5.1 Dielectrics:

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Section 6: Electromagnetic Waves #

Electromagnetic Waves

Section 6:

6.1 Wave Equation and Poynting Vector #

6.1 Wave Equation and Poynting Vector

Wave Equations Derivation:

Start by taking the culr of: $ \nabla \times \mathbf{E} = - \frac{\partial{\mathbf{B}}}{\partial{t}} $.
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Explicit Wave Solutions:

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Poynting Vector:

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Intensity:

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Electromagnetic Waves at Boundaries:

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Also $ \mathbf{E_{inc}} + \mathbf{E_{refl}} = E_{transmitted} $ and in general: $ \mathbf{E}^{||}_{out} = \mathbf{E}^{||}_{in} $.

6.2 Radiation #

6.2 Radiation

Larmor Formula:

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Dipole Radiation:

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Magnetic Dipole Radiation:

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Monopoles do NOT Radiate:

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Section 7: Circuits #

7. Circuits

Section 7

7.1 Basic Elements #

7.2 Kirchhoff's Rules #

7.3 Energy in Circuits #

7.4 Standard Circuit Types #

7.1 Basic Elements

7.2 Kirchhoff's Rules:

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7.3 Energy in Circuits:

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7.4 Standard Circuit Types

Voltage across each Element:

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Elements in Parallel:

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Elements in Series:

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Resistance:

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RL Circuits:

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RC Circuits:

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RLC Circuits:

Transient and Steady-state behavior:

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Like Currents Attract

Introduction

One point of physics should be mentioned. Historically, electrostatics developed as a science of macroscopic phenomena. As indicated at the end of the Introduction, such idealizations as point charges or electric fields at a point must be viewed as mathematical constructs that permit a description of the phenomena at the macroscopic level, but that may fail to have meaning microscopically.

1.1 Coulomb's Law #

1.2 Electric Field #

1.3 Gauss's Law #

1.4 Differential Form of Gauss's Law #

1.5 Another Equation of Electrostatics and the Scalar Potential #

1.6 Surface Distributions of Charges and Dipoles and Discontinuities in the Electric Field and Potential #

1.7 Poisson and Laplace Equations #

1.8 Green's Theorem #

1.9 Uniqueness of the Solution with Dirichlet or Neumann Boundary Conditions #

1.10 Formal Solution of Electrostatic Boundary-Value Problem with Green Function #

1.11

1.12

1.13

Electric Field

The electric field can be defined as the force per unit charge acting at a given point. It is a vector function of position, denoted by $ \mathbf{E} $.

Definition

In symbols we may write: $ \mathbf{F} = q\mathbf{E} $ where F is the force, E the electric field, and q the charge. In this equation, it is assumed that the charge q is located at a point, and the force and the electric field are evaluated at that point.

Point Charge

System of Point Charges

Continuous Distribution

A discrete set of point charges can be described with a charge density by means of delta functions. For example:
$ ρ(\mathbf{x}) = \sum^n_i q_i δ(\mathbf{x} - \mathbf{x_ι}) $
represents a distribution of $ n $ point charges $ q_i $, located at the points $ \mathbf{x_i} $.

Note: One must be careful in its definition, however. It is not necessarily the force that one would observe by placing one unit of charge on a pith ball and placing it in position. The reason is that one unit of charge may be so large that its presence alters appreciably the field configuration of the array. Consequently one must use a limiting process whereby the ratio of the force on the small test body to the charge on it is measured for smaller and smaller amounts of charge. Experimentally, this ratio and the direction of the force will become constant as the amount of test charge is made smaller and smaller. These limiting values of magnitude and direction define the magnitude and direction of the electric field $ \mathbf{E} $ at the point in question.

Note: The constant k differs in different systems of units.

The experimentally observed linear superposition of forces due to many charges means that we may write the electric field at $ \mathbf{x} $ due to a system of point charges $ q_i $, located at $ \mathbf{x} $, $ i = 1, 2, ldots , n $, as the vector sum:
$ \mathbf{E(x)} = \frac{1}{4 π ε_0} \sum^n_i q_i \frac{\mathbf{x} - \mathbf{x_i}}{|\mathbf{x} - \mathbf{x_i}|^3} $

If the charges are so small and so numerous that they can be described by a charge density $ ρ(x') $, the sum is replaced by an integral:
$ \mathbf{E(x)} = \frac{1}{4 π ε_0} \int ρ(x') \frac{\mathbf{x} - \mathbf{x'}}{|\mathbf{x} - \mathbf{x'}|^3} d^3x'$,
where $ d^3x' = dx' dy' dz' $ is a three-dimensional volume element at $ \mathbf{x'} $.

Gauss's Law

Element of Solid Angle:

$ dΩ = \frac{cosθ}{r^2} \, dα $

For a discrete set of charges:
$ \oint_S \mathbf{E} . \mathbf{n} \, da = \frac{1}{ε_0} \sum_i q_i $
where the sum is over only those charges inside the surface S.

For a continuous
charge density p(x), Gauss’s law becomes:
$ \oint_S \mathbf{E} . \mathbf{n} \, da = \frac{1}{ε_0} \int_V ρ(\mathbf{x}) d^3x $
where V is the volume enclosed by S.

Equation (#) is one of the basic equations of electrostatics. Note that it depends upon:

the inverse square law for the force between charges,
the central nature of the force, and
the linear superposition of the effects of different charges.

Differential Form of Gauss's Law

$ \mathbf{\nabla} . \mathbf{Ε} = ρ / ε_0 $

Another Equation of Electrostatics and the Scalar Potential

Necessity of Curl $ \mathbf{E} $

Proof

Using:
$ \frac{\mathbf{x} - \mathbf{x'}}{|\mathbf{x} - \mathbf{x'}|^3} = - \nabla \left( \frac{1}{|\mathbf{x} - \mathbf{x'}|}\right ) $,
we can change:
$ \mathbf{E(x)} = \frac{1}{4 π ε_0} \int ρ(\mathbf{x'}) \frac{\mathbf{x} - \mathbf{x'}}{|\mathbf{x} - \mathbf{x'}|^3} \, d^3x'$
$ = -\frac{1}{4 π ε_0} \int ρ(\mathbf{x'}) \nabla\left( \frac{1}{|\mathbf{x} - \mathbf{x'}|} \right) d^3x'$
Since the gradient operation involves $ \mathbf{x} $, but not the integration variable $ \mathbf{x'} $, it can be taken outside the integral sign. Then the field can be written:
$ = -\frac{1}{4 π ε_0} \nabla \left( \int ρ(\mathbf{x'}) \frac{1}{|\mathbf{x} - \mathbf{x'}|} \; d^3x' \right) $
Since the curl of the gradient of any well-behaved scalar function of position
vanishes, then it follows immediately that: $ \boxed{\nabla \times \mathbf{E} = 0 } $.

The single equation $ \mathbf{\nabla} . \mathbf{Ε} = ρ / ε_0 $ is not enough to specify completely the three components of the electric field E(x). A vector field can be specified almost completely (up to the gradient of a scalar function that satisfies the Laplace equation), if its divergence and curl are given everywhere in space.

Note that $ \nabla x \mathbf{E} = 0 $ depends on the central nature of the force between charges, and on the fact that the force is a function of relative distances only, but does not depend on the inverse square nature.

Scalar Potential

$ \mathbf{E} = -\nabla Φ $

$ Φ(\mathbf{x}) = \frac{1}{4 π ε_0} \int \frac{ρ(\mathbf{x'})}{|\mathbf{x} - \mathbf{x'}|} \; d^3x' $

$ q Φ $ can be interpreted as the potential energy of a test charge $ q $ in the electrostatic field.

Line integral of $ \mathbf{E} $

The line integral of the electric field
between two points is independent of the path and is the negative of the potential
difference between the points:
$ \int_A^B \mathbf{E} . d\mathbf{l} = -(Φ_B - Φ_A) $.

If the path is closed, the line integral is zero:
$ \oint \mathbf{E} . d\mathbf{l} = 0 $, a result that can also be obtained directly from Coulomb’s law.

Surface Distributions of Charges and Dipoles and Discontinuities in the Electric Field and Potential

Continuity of the Potential

For volume or surface distributions of charge, the potential, is everywhere continuous, even within the charge distribution. This can be shown from the fact that E is bounded, even though discontinuous across a surface distribution of charge. With point or line charges, or dipole layers, the potential is no longer continuous.

Dipole Surface Distribution

Potential

Potential Discontinuity

Surface Dipole Moment

Poisson and Laplace Equations

Identity

Green's Theorem

Green's First Identity

Green's Second Identity or Theorem

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Potential Equation

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Uniqueness of the Solution with Dirichlet or Neumann Boundary Conditions

Boundary Conditions for unique solutions

What boundary conditions are appropriate for the Poisson (or Laplace) equation to ensure that a unique and well-behaved (i.e., physically reasonable) solution will exist inside the bounded region? Physical experience leads us to believe that specification of the potential on a closed surface (e.g., a system of conductors held at different potentials) defines a unique potential problem. This is called a Dirichlet problem, or Dirichlet boundary conditions. Similarly it is plausible that specification of the electric field (normal derivative of the potential) everywhere on the surface (corresponding to a given surface-charge density) also defines a unique problem. Specification of the normal derivative is known as the Neumann boundary condition.

Only one Kind of Boundary Conditions

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Formal Solution of Electrostatic Boundary-Value Problem with Green Function

Definition of Green's Function

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General Potential Solution

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Dirichlet Boundary Conditions

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Neumann Boundary Conditions

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Mathematical Property

$ G(\mathbf{x},\mathbf{x'}) = G(\mathbf{x'},\mathbf{x}) $

Physical Meaning of $ F(\mathbf{x},\mathbf{x'}) $

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Ch. 7 : Plane Electromagnetic Waves and Wave Propagation #

Ch. 8 : Waveguides, Resonant Cavities, and Optical Fibers

Ch. 9 : Radiating Systems, Multipole Fields and Radiation

Ch. 10 : Scattering and Diffraction

Ch. 11 : Special Theory of Relativity

Ch. 12 : Dynamics of Relativistic Particles and Electromagnetic Fields

Ch. 13 : Collisions, Energy Loss, and Scattering of Charged Particles, Cherenkov and Transition Radiation

Ch. 14 : Radiation by Moving Charges

Ch. 15 : Bremsstrahlung, Method of Virtual Quanta, Radiative Beta Processes

Ch. 16 : Radiation Damping, Classical Models of Charged Particles

Plane Electromagnetic Waves and Wave Propagation

Chapter 7

7.1 Plane Waves in a Nonconducting Medium #

7.2 Linear and Circular Polarization; Stokes Parameters

7.3 Reflection and Refraction of Electromagnetic Waves at a Plane Interface Between Two Dielectrics

7.4 Polarization by Reflection; Total Internal Reflection; Goos-Hanchen Effect

7.5 Frequency Dispersion Characteristics of Dielectrics, Conductors, and Plasmas #

7.6 Simplified Model of Propagation in the ionosphere and Magnetosphere

7.7 Magnetohydrodynamic Waves

7.8 Superposition of Waves in One Dimension; Group Velocity

7.9 Illustration of the Spreading of a Pulse As It Propagates in a Dispersive Medium

7.10 Causality in the Connection Between D and E; Kramers-Kronig Relations #

7.11 Arrival of a Signal After Propagation Through a Dispersive Medium