Modern Electrodynamics

The second is my use of the imaginary numerito to impose the metric in special relativity. At the top of the list, I gratefully acknowledge the contributions of Wayne Saslow (Texas A&M University) and Glenn Smith (Georgia Tech).

Introduction

Vectors

Cartesian Coordinates
Cylindrical Coordinates
Spherical Coordinates
The Einstein Summation Convention
The Kronecker and Levi-Civit`a Symbols
Vector Identities in Cartesian Components
Vector Identities in Curvilinear Components

The volume element in spherical coordinates isd3r=r2sinθdrdθdφ. The unit vectors are related by. The identity on the left-hand side of (1.37) and the invariance of the Levi-Civit`a symbol with respect to cyclic permutations of its indices gives.

Figure 1.1: An orthonormal set of unit vectors ˆ e 1 , ˆ e 2 , ˆ e 3 . V is an arbitrary vector.

Two Identities for ∇ × L

Derivatives

Functions of r and | r |
Functions of r − r
The Convective Derivative
Taylor’s Theorem

Integrals

Jacobian Determinant
The Divergence Theorem
Green’s Identities
Stokes’ Theorem
The Time Derivative of a Flux Integral

Generalized Functions

The Delta Function in One Dimension
The Principal Value Integral and Plemelj Formula
The Step Function and Sign Function
The Delta Function in Three Dimensions
Some Useful Delta Function Identities

Fourier Analysis

Parseval’s Theorem
The Convolution Theorem
A Time-Averaging Theorem

Orthogonal Transformations

Passive Point of View
Active Point of View

Cartesian Tensors

Reflection, Inversion and Pseudotensors

Our treatment of Faraday's law uses the time derivative of the surface integral, where the surface S(t) moves because its individual surface elements move with velocityυ(r, t). In curvilinear coordinates, this is indicated by the restriction on the right-hand side of (1.113) and the shape of the volume elements for cylindrical and spherical coordinates.

Figure 1.3: A surface S(t) [bounded by the solid curve labeled C(t)] changes to the surface S(t + δt) (dashed curve) because each element of surface moves by an amount υ δt.

Inversion and Reflection

The Helmholtz Theorem

This result is valid for both static and time-dependent vector fields. 1.176) Substituting C(r) and∇2 in the last term and using the double curl identity in Example 1.3 gives. This follows from (1.76) and (1.78), which show that the divergence theorem transforms the volume integrals in the last two terms of (1.181) into the surface integrals.

Lagrange Multipliers

Sources, References, and Additional Reading

Problems

Levi-Civit`a Practice I

Levi-Civit`a Practice II Evaluate the following expressions which exploit the Einstein summation con- vention

Delta Function Identities A test function as part of the integrand is required to prove any delta function identity. With this in mind,

An Application of Stokes’ Theorem Without using vector identities, (a) use Stokes’ Theorem

Compute the Normal Vector Compute the unit normal vector ˆ n to the ellipsoidal surfaces defined by constant values of

A Variant of the Helmholtz Theorem I Mimic the proof of Helmholtz’ theorem in the text and prove that

Dot and Cross Products Let b be a vector and ˆn a unit vector

If you wake up a physicist in the middle of the night and say "Maxwell", he is sure to say "electromagnetic field".

Introduction

Electric Charge
Electric Current
Conservation of Charge

For our purposes, the most useful statement of charge conservation begins with the surface-integral representation of the current Iin (2.7). The continuity equation states that the total charge in any infinitesimal volume is constant unless there is a net flow of pre-existing charge into or out of the volume through its surface.

Figure 2.1: (a) The current I is the integral over S of the projection of the volume current density j onto the area element dS; (b) the current I is the integral over C of the projection of K × nˆ (the cross product of the surface current density K and th

Moving Point Charges

The Maxwell Equations in Vacuum

Electrostatics
The Field Concept
Magnetostatics
Faraday’s Law
The Displacement Current
Putting It All Together

Microscopic vs. Macroscopic

Lorentz Averaging
The Macroscopic Surface
Matching Conditions

The Maxwell Equations in Matter

Macroscopic Sources and Fields
Microscopic Fields

Quantum Limits and New Physics

Quantized Matter
Vacuum Polarization
Quantum Fluctuations
New Physics
Magnetic Charge

SI Units
A Heuristic Derivation
The Coulomb and Biot-Savart Laws The electric and magnetic fields for time-independent distributions of charge and current which go to zero at infinity are
Time-Dependent Charges at Rest Consider a collection of point particles fixed in space with charge density ρ(r, t ) = !
A Variation of Coulomb’s Law Suppose that the electric potential of a point charge at the origin were ϕ(r) = q
Introduction

The Scope of Electrostatics

Coulomb’s Law
The Scalar Potential

The Coulomb Force is Conservative
Matching Conditions for ϕ(r)
Earnshaw’s Theorem
Equipotential Surfaces and Electric Field Lines
A Charged Line Segment

Gauss’ Law and Solid Angle

The Importance of Symmetry
Matching Conditions for E(r)
The Force on a Charged Surface
Solid Angle

Maxwell characterized his theory as an attempt to "mathematize" the results of many different experimental investigations of electrical and magnetic phenomena. Another characteristic feature—the "spillout" of the electron distribution in the vacuum—becomes most apparent when we average ρ0(r) over planes parallel to . If the unit normal ˆn1 points outward from region 1 and the unit normal ˆn2 points outward from region 2 (Figure 2.8), the full set of matching conditions.

Maxwell's equations are a mathematical expression of known facts of experimental electromagnetism. Finally, SI eliminates ("rationalizes") the 4π factors in Maxwell's equations by introducing the magnetic constant μ0where. Example 3.2 The charge distributions in different atomic nuclei look very similar except for the change in extent.3 In Figure 3.2, ρA(r) is the charge density of the A nucleus and ifρB(r) is the charge density of the B nucleus.

The macroscopic adaptation conditions for the electrostatic potential follow from the behavior of the electric field near an interface with surface charge densityσ(rS) (Section 2.3.3). 68 ELECTROSTATICS: THE ELECTRIC FIELD PRODUCED BY STATIONARY CHARGE . the hole) with a surface charge density equal to the negative surface charge density of the shell.

Figure 2.2: Two filamentary loops carry current I and I k . The vectors r and r k point to the line elements d and d k , respectively.

Field Lines for a Point Charge in a Uniform Field

Electrostatic Potential Energy

Coulomb Force from Variation of Potential Energy
Green’s Reciprocity Relation

Electrostatic Total Energy

U E is Positive-Definite
Interaction Total Energy is Potential Energy

It states that the potential energy of ρ2(r) in the field produced by ρ1(r) is equal to the potential energy of ρ1(r) in the field produced by ρ2(r). Instead of evaluating (3.76), we can calculate the work required to "build" the ball by successively adding spherical layers of uniform charge and thickness. A variation of the method used to obtain (3.77) can be used to derive (3.76) without going through the point load result (3.75).

3.78) Finally, it is worth noting that the total electrostatic energy UE is a quantity that clearly illustrates the fundamental limitations of the point charge concept in classical electrodynamics.UE. If qk= |rk| =1, numerical studies show that the energy of the minimum energy configuration is very well approximated by. Roughly, the first term is the average energy of N charges randomly distributed over the surface of the sphere.

Figure 3.13: Demonstration that δρ(r) = ρ(r − δs) − ρ(r).

The Ionization Potential of a Metal Cluster

The Electric Stress Tensor

Applications of the Electric Stress Tensor

Charged Particle Refraction
Symmetric and Traceless The Cartesian components of the electric field in a charge-free region of space are E k = C k + D j k r j , where C k and D j k are constants
Practice Superposing Fields This problem exploits the ring and disk electric fields calculated in Example 2.1
Five Charges in a Line Draw the electric field line pattern for a line of five equally spaced charges with equal magnitude but alternating algebraic signs, as sketched below
Gauss’ Law Practice Use Gauss’ law to find the electric field when the charge density is
The Electric Field of a Charged Slab and a Charged Sheet (b) Find E(r) if ρ(x, y, z) = σ 0 δ(x) + ρ 0 θ(x) − ρ 0 θ(x − b)
Two Electrostatic Theorems (a) Use Green’s second identity,
Interaction Energy of Spheres
Electrostatic Interaction Energy
Ionization Energy of a Model Hydrogen Atom A model hydrogen atom is composed of a point nucleus with charge +| e | and an electron charge distribution
Two Electric Field Formulae
Introduction

The Electric Multipole Expansion

The Electric Dipole

The Point Electric Dipole
The Singularity at the Origin
The Dipole Force
The Dipole Torque
The Dipole Potential Energy
The Dipole-Dipole Interaction

Electric Dipole Layers

The Potential of a Dipole Layer
Matching Conditions at a Dipole Layer

The electron is modeled as a negative point charge in the center of the sphere (black dot). With regard to the position of the electron, Gauss's law gives the electric field of the sphere as . To find the force on the charge atz= −d, we choose as the surface area of the massive cup-shaped object in Figure 3-18.

If both charges are counted, the component of the electric field is in the plane of the center line at point P. Using a side (edge) view, sketch the electric field lines everywhere assuming that the total charge of the disc is positive is. The matching condition,0nˆ2·[E1−E2]=σ(rS), tells us that the second term in (4.41) generates a jump in the normal component of the electric field atz=0.

Figure 3.15 shows models for the cluster before and after the ionization event. The initial state diagram shows the atomic sphere of the one atom that will lose its valence electron by ionization

Monolayer Electric Dipole Drops

The Electric Quadrupole

The Traceless Quadrupole Tensor
Force and Torque on a Quadrupole

The discontinuity (4.45) implies that the tangential component of the electric field is not continuous if the dipole layer density varies along the surface. The width w0 is independent of the droplet lengthL, as claimed in the problem statement. Repeating (4.6), the components of the electric quadrupole tensor are Qij Mimicking (4.9), it is not difficult to show that these scalars are uniquely defined and do not depend on the choice of the origin of coordinates if Q=p=0 .

9 The multiple derivative structure in (4.3) guarantees that each term in the multipole expansion can be represented by point charge configurations constructed from suitable pairings in space of the configurations describing the previous term. Therefore, a complete characterization of the quadrupole potential in an arbitrary coordinate system requires the specification of only five numbers (the independent components of the traceless Cartesian quadrupole tensor) instead of six numbers (the components of the primitive Cartesian quadrupole tensorQ). Following the logic of Section 4.2.3, it is not difficult to show that and (4.69) are valid for any neutral charge distribution with p=0 if the electric field does not vary too rapidly over the volume of the distribution.

Nuclear Quadrupole Moments

Spherical Mathematics

Legendre Polynomials
Spherical Harmonics
The Inverse Distance

Spherical and Azimuthal Multipoles

The Exterior Spherical Expansion
The Interior Spherical Expansion
Azimuthal Multipoles
Preview: the Connection to Potential Theory

It would be much more convenient if the right-hand side of (4.78) were expressed explicitly in terms of spherical coordinates r=(r, θ, φ) andr=(r, θ, φ) defined with respect to a certain set of coordinates. axes in space. We derive multipole expansions of the electrostatic potential ϕ(r) suitable for systems with natural spherical symmetry by substituting (4.84) into When the charge density is localized and confined to the sphere of radiusRin Figure 4.1, the spherical analogue of the Cartesian expansion (4.3) which converges for observation points outside the sphere is. 4.86).

The Cartesian analogue of the inner expansion (4.89) is a power expansion of ϕ(x, y, z) around the origin of the coordinates. We need two properties for the spherical harmonics: 4.94) Substituting A mandB minto (4.86) and (4.89), respectively, and using (4.91) gives exterior and interior azimuthal multipole expansion in the form. This reflects the continuity of the electrostatic potential as the observation point passes through a charge layer.

Figure 4.13: Two spherical charge distributions that do not overlap in space.

The Liquid Drop Model of Nuclear Fission

Primitive and Traceless Multipole Moments

Counting Multipole Moments

Assertion B: If U0 is the self-energy of the undistorted nucleus, the change in electrostatic energy caused by distortion of the form (4.101) is. There is a deep connection between the spherical multipole expansion derived in Section 4.6 and a traceless version of the Cartesian multiple expansion derived in Section 4.1.1, namely that the number of independent Cartesian multipole moments without trace Tij ..m() is equal to the number of spherical multipole momentsAmStarting at=2, the number in question is smaller (often much smaller) than the number of primitive independent Cartesian multipole moments,Cij ..m(). However, the term=2 is not the quadrupole potential (4.52) expressed in terms of primitive momentsQij.

4.125) The representation of the electrostatic potential by this traceless Cartesian expansion (4.123) is completely equivalent to the primitive Cartesian expansion (4.119). This means that the number of independent components of the tensor C()—call this number M—is equal to the number of ways in which each index takes one of the three values (x, y, z), regardless of their order. In the language of group theory, both representations of the potential are irreducible, that is, maximally efficient in our language.

Figure 4.17: Small-amplitude distortions of a unit sphere: (a) R(θ) = α 1 P 1 (cos θ);

The Dielectric Polarization P(r)

Dipole Moment Practice Find the electric dipole moment of
Smolochowski’s Model of a Metal Surface A surprisingly realistic microscopic model for the charge density of a semi-infinite metal (with z = 0 as its macroscopic surface) consists of a positive charge
The Charge Density of a Point Electric Dipole The text used Poisson’s equation to show that the charge density of a point electric dipole with moment p located at the point r 0 is ρ D (r) = − p · ∇ δ(r − r 0 )
Stress Tensor Proof of No Self-Force Use the electric stress tensor formalism to prove that no isolated charge distribution ρ(r) can exert a net force on itself. Distinguish the cases when ρ(r) has a net charge and
Potential of a Double Layer
Interaction Energy of Adsorbed Molecules Molecules adsorbed on the surface of a solid crystal sur- face at low temperature typically arrange themselves into a periodic arrangement, e.g., one molecule lies
The Many Faces of a Quadrupole
Properties of a Point Electric Quadrupole
Interaction Energy of Nitrogen Molecules How does the leading contribution to the electrostatic inter- action energy between two nitrogen molecules depend on the distance R between them?
Foldy’s Formula The low-energy Born approximation to the amplitude for electron scattering from a neutron is proportional to the volume integral of the potential energy of interaction between the electron and
Practice with Spherical Multipoles
Exterior Multipoles for Specified Potential on a Sphere
Analyze This Potential An asymptotic (long-distance) electrostatic potential has the form ϕ(r, θ, φ) = A
Introduction
Electrostatic Induction

Thomson’s Theorem of Electrostatics
The Surface Charge Density

Screening and Shielding
Capacitance

Self-Capacitance
What Does it Mean to ‘‘Ground’’ a Conductor?
The Capacitance Matrix
The Two-Conductor Capacitor

The Energy of a System of Conductors

Pollock, "The calculation of the electrostatic energy in the liquid drop model of nuclear fission: A pedagogical note", Physica A. Find the angle α that minimizes the total energy and show that the energy/dipole is U= 1. b) The N2 molecule has a small quadrupole moment because covalent bonding builds up a negative (electron) charge in the bonding region between the atomic nuclei. Measurements of the electrostatic potential outside the box show that all the external multipole moments for = 1,2,. the electrostatic potential outside the box shows that all the external multipole moments for = 1,2,. is zero in a coordinate system with its origin at the center of the box.

Use the results of part (a) to find the asymptotic (r→ ∞) form of the potential produced by the shell. If there is source charge outside the conductor, the outer surface of the conductor develops a surface charge density. The conductor develops a surface charge density on the surface of the cavity and on the outer surface of the conductor.

Figure 5.1: Electrostatic induction in a neutral conductor. (a) External field E ext (r) induces charge rearrangement; (b) The charge distribution that produces mechanical equilibrium creates a field E self (r) (not shown) with the property that E(r) = E e

Coulomb Blockade

Forces on Conductors

Charges Held Constant
Potentials Held Constant
Why U E Differs From U ˆ E

Real Conductors

Debye-H¨uckel and Thomas-Fermi Screening

A Dipole in a Cavity A point electric dipole with moment p is placed at the center of a hollow spherical cavity scooped out of an infinite conducting medium
A Conducting Disk
The Capacitance of Spheres
Maxwell Was Not Always Right A non-conducting square has a fixed surface charge distribution. Make a rectangle with the same area and total charge by cutting off a slice from one side of the square and gluing
Force Equivalence Confirm the assertion made in the text that the inverse relation between the matrix of capacitive coefficients and the matrix of potential coefficients implies the equivalence of these two
Introduction
Polarization

The Volume Integral of P(r)
The Lorentz Model
The Modern Theory of Polarization

The Field Produced by Polarized Matter

Also, there is no contribution to the force formula from the energy stored in the capacitor to the right of pointx+. The electric field is zero in the main part of the conductor (grey) except within a shielding layer (black) of thickness on the surfaces. Marcus, "The theory of the triode as a three-body problem in electrostatics", The American Physics Teacher.

Find the capacitance elements with potential difference ϕ2−ϕ1>0 as shown in the diagram below. To get a flavor of this approach, we recall from (6.4) that the dielectric charge density ρ(r) is the (negative) polarization divergence. However, if the polarization were simply the dipole moment per unit volume of charge density, P(r) would contain less information than ρ(r).

Figure 5.12: Left: two-dimensional quantum dot (shaded circle) separated from an electron reservoir (shaded annulus) by an insulating ring (white)

A Uniformly Polarized Sphere

Polarized Matter as a Superposition of Point Dipoles
The Total Electric Field

The Auxiliary Field D(r)
Matching Conditions
Constitutive Relations

Simple Dielectric Matter

Fields and Sources in Simple Dielectric Matter
Simple Dielectric Response to Free Charge
Polarization Charge at a Simple Interface
Simple Dielectric Response to Fixed Fields

The point dipole representation of the polarization qualitatively rationalizes the dependence of the polarization charge densities ρP(r) and σP(r) on P(r). The total electric field E in the Maxwell equations is the sum of the electric field EP produced by the polarization chargeρP(r)= −∇ ·Pand the electric field produced by all charges not associated with dielectric bodies. Gauss's law concerns the total charge, which is the sum of the polarization charge and the free charge.

Following the method of Section 2.3.3 (or otherwise), the matching conditions implied by (6.27) and (6.28) at an interface between two regions of space endowed with a surface density of free chargeσf. The electric field is reduced (shielded) compared to its value E0=zσ/ˆ 0 in the absence of dielectric. In other words, E=E0 when the dielectric is present because the charge flows from the battery (or whatever keeps the plates at a fixed potential) to the surface of the plates to exactly cancel the polarization charge on the adjacent dielectric surfaces.

Figure 6.4: One-dimensional point dipole representation of the polarization P(r) with ρ P = −∇ · P and σ P = P · n

Refraction of Field Lines at a Dielectric Interface

Potential Theory for a Simple Dielectric
The Physics of the Dielectric Constant

The Electric Polarizability
The Clausius-Mossotti Formula
Polar Liquids and Solids
The Limit κ → ∞

The Energy of Dielectric Matter

If so, there must also be regions within the dielectric where the electric fields are smaller than (or opposite in direction to) the field of the vacuum capacitor. This is consistent with the discussion immediately above of a parallel plate capacitor with a fixed plate potential. The Lorentz model for polarizable matter (Section 6.2.2) relates the macroscopic dielectric constant κ [see (6.36)] of the dielectric gas to the microscopic polarizability α [see (6.75)] of the constituent atoms or molecules.

We also need an alternative definition of the cell-averaged dipole moment pR first defined in (6.10), namely,. On the other hand, the macroscopic fieldE(R) in (6.78) is the average of the field in a given cell from all sources, incl. This formula confirms the remark made at the end of Section 6.5.4 that the local electric field typically exceeds the average macroscopic field within a dielectric. Elocal(R) is due to all teoter molecules, so it is characteristic of the volume between gas molecules.