- •Contents
- •Preface
- •How to use this book
- •Chapter 1 Units, constants, and conversions
- •1.1 Introduction
- •1.2 SI units
- •1.3 Physical constants
- •1.4 Converting between units
- •1.5 Dimensions
- •1.6 Miscellaneous
- •Chapter 2 Mathematics
- •2.1 Notation
- •2.2 Vectors and matrices
- •2.3 Series, summations, and progressions
- •2.5 Trigonometric and hyperbolic formulas
- •2.6 Mensuration
- •2.8 Integration
- •2.9 Special functions and polynomials
- •2.12 Laplace transforms
- •2.13 Probability and statistics
- •2.14 Numerical methods
- •Chapter 3 Dynamics and mechanics
- •3.1 Introduction
- •3.3 Gravitation
- •3.5 Rigid body dynamics
- •3.7 Generalised dynamics
- •3.8 Elasticity
- •Chapter 4 Quantum physics
- •4.1 Introduction
- •4.3 Wave mechanics
- •4.4 Hydrogenic atoms
- •4.5 Angular momentum
- •4.6 Perturbation theory
- •4.7 High energy and nuclear physics
- •Chapter 5 Thermodynamics
- •5.1 Introduction
- •5.2 Classical thermodynamics
- •5.3 Gas laws
- •5.5 Statistical thermodynamics
- •5.7 Radiation processes
- •Chapter 6 Solid state physics
- •6.1 Introduction
- •6.2 Periodic table
- •6.4 Lattice dynamics
- •6.5 Electrons in solids
- •Chapter 7 Electromagnetism
- •7.1 Introduction
- •7.4 Fields associated with media
- •7.5 Force, torque, and energy
- •7.6 LCR circuits
- •7.7 Transmission lines and waveguides
- •7.8 Waves in and out of media
- •7.9 Plasma physics
- •Chapter 8 Optics
- •8.1 Introduction
- •8.5 Geometrical optics
- •8.6 Polarisation
- •8.7 Coherence (scalar theory)
- •8.8 Line radiation
- •Chapter 9 Astrophysics
- •9.1 Introduction
- •9.3 Coordinate transformations (astronomical)
- •9.4 Observational astrophysics
- •9.5 Stellar evolution
- •9.6 Cosmology
- •Index
142 |
Electromagnetism |
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7.4 Fields associated with media
Polarisation
Definition of electric |
p = qa |
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(7.80) |
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dipole moment |
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Generalised electric |
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ρ dτ |
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dipole moment |
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volume |
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Electric dipole |
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p · r |
(7.82) |
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potential |
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4π 0r3 |
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Dipole moment per |
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unit volume |
P = np |
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(7.83) |
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(polarisation)a |
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Induced volume |
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charge density |
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Induced surface |
σind = P · sˆ |
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(7.85) |
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charge density |
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Definition of electric |
D = 0E + P |
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displacement |
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Definition of electric |
P = 0χE E |
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(7.87) |
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susceptibility |
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Definition of relative |
r = 1 + χE |
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D = 0 rE |
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permittivityb |
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Atomic |
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polarisabilityc |
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Depolarising fields |
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NdP |
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E d = − 0 |
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Clausius–Mossotti |
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equation |
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3 0 |
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r + 2 |
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±q end charges
acharge separation vector (from − to +)
pdipole moment
ρcharge density
dτ |
volume element |
r |
vector to dτ |
φdipole potential
rvector from dipole
0 |
permittivity of free |
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space |
Ppolarisation
nnumber of dipoles per unit volume
ρind |
volume charge density |
σind |
surface charge density |
sˆ |
unit normal to surface |
Delectric displacement
Eelectric field
χE |
electrical susceptibility |
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(may be a tensor) |
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relative permittivity |
permittivity
αpolarisability
E loc |
local electric field |
E d |
depolarising field |
Nd |
depolarising factor |
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cylinder, axis to P ) |
− p +
aAssuming dipoles are parallel. The equivalent of Equation (7.112) holds for a hot gas of electric dipoles. bRelative permittivity as defined here is for a linear isotropic medium.
cThe polarisability of a conducting sphere radius a is α = 4π 0a3. The definition p = α 0E loc is also used.
dWith the substitution η2 = r [cf. Equation (7.195) with µr = 1] this is also known as the “Lorentz–Lorenz formula.”
7.4 Fields associated with media |
143 |
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Magnetisation
Definition of |
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magnetic dipole |
dm = I ds |
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moment |
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Generalised |
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magnetic dipole |
m = |
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(7.95) |
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moment |
volume |
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Magnetic dipole |
φm(r) = |
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(7.96) |
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(scalar) potential |
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4πr |
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Dipole moment per |
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unit volume |
M = nm |
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(7.97) |
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(magnetisation)a |
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Induced volume |
J ind = ×M |
(7.98) |
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current density |
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Induced surface |
j ind = M ×sˆ |
(7.99) |
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current density |
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Definition of |
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magnetic field |
B = µ0(H + M ) |
(7.100) |
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strength, H |
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M = χH H |
(7.101) |
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Definition of |
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χBB |
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magnetic |
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µ0 |
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susceptibility |
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χH |
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χB = |
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B = µ0µrH |
(7.104) |
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Definition of relative |
= µH |
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(7.105) |
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µr = 1 + χH |
(7.106) |
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permeabilityb |
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1 − χB |
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dm dipole moment
Iloop current
ds loop area (right-hand sense with respect to loop current)
mdipole moment
Jcurrent density
dτ |
volume element |
r |
vector to dτ |
φm |
magnetic scalar |
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potential |
rvector from dipole
µ0 |
permeability of free |
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Mmagnetisation
nnumber of dipoles per unit volume
J ind |
volume current |
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density (i.e., A m−2) |
j ind |
surface current |
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density (i.e., A m−1) |
sˆ |
unit normal to |
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surface |
Bmagnetic flux density
Hmagnetic field strength
χH magnetic susceptibility. χB is also used (both may be tensors)
µr |
relative permeability |
µpermeability
dm, ds
outin
7
aAssuming all the dipoles are parallel. See Equation (7.112) for a classical paramagnetic gas and page 101 for the quantum generalisation.
bRelative permeability as defined here is for a linear isotropic medium.
144 Electromagnetism
Paramagnetism and diamagnetism
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m |
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magnetic moment |
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r2 |
mean squared orbital radius |
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Diamagnetic |
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e2 |
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(of all electrons) |
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2 |
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Z |
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atomic number |
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moment of an atom |
m = − |
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Z r |
B |
(7.108) |
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6me |
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B |
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magnetic flux density |
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me |
electron mass |
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−e |
electronic charge |
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Intrinsic electron |
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e |
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J |
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total angular momentum |
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m − |
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magnetic momenta |
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Lande´ g-factor (=2 for spin, |
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2me |
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=1 for orbital momentum) |
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1 |
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Langevin function |
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Langevin function |
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Classical gas |
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m0B |
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M |
apparent magnetisation |
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paramagnetism |
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m0 |
magnitude of magnetic dipole |
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M = nm0L kT |
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moment |
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dipole number density |
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µ0nm02 |
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T |
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temperature |
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Curie’s law |
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Boltzmann constant |
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χH = |
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χH |
magnetic susceptibility |
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Curie–Weiss law |
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permeability of free space |
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Curie temperature |
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aSee also page 100. |
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Boundary conditions for E , D, B, and H a
Parallel |
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electric field |
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interface |
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Perpendicular |
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density |
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interface |
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D1,2 |
electrical displacements |
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in media 1 & 2 |
Electric |
sˆ · (D2 − D1) = σ |
(7.117) |
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unit normal to surface, |
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directed 1 → 2 |
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surface density of free |
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charge |
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H 1,2 |
magnetic field strengths |
Magnetic field |
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sˆ×(H 2 − H 1) = j s |
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j s |
surface current per unit |
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width |
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aAt the plane surface between two uniform media. bIf σ = 0, then D is continuous.
cIf j s = 0 then H is continuous.
2
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7.5 Force, torque, and energy |
145 |
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7.5 Force, torque, and energy
Electromagnetic force and torque
Force between two |
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q1q2 |
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static charges: |
F 2 = |
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Force between two |
dF 2 = |
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elements |
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Force on a |
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current-carrying |
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Force on a charge |
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Force on an electric |
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Torque on an |
G = p×E |
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electric dipole |
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Torque on a |
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magnetic dipole |
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Torque on a |
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current loop |
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F 2 |
force on q2 |
q1,2 |
charges |
r12 |
vector from 1 to 2 |
ˆunit vector
0 |
permittivity of free |
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dl1,2 |
line elements |
I1,2 |
currents flowing along |
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dF 2 |
force on dl2 |
µ0 |
permeability of free |
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dl |
line element |
Fforce
I current flowing along dl
Bmagnetic flux density
Eelectric field
vcharge velocity
pelectric dipole moment
mmagnetic dipole moment
Gtorque
dlL |
line-element (of loop) |
rposition vector of dlL
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current around loop |
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simplifies to (p · E ) if p is intrinsic, (pE/2) if p is induced by E and the medium is isotropic. |
bF |
F simplifies to (m · B) if m is intrinsic, (mB/2) if m is induced by B and the medium is isotropic.
F 2
q1 r12 q2
dl1
r12
dl2
7
146 |
Electromagnetism |
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Electromagnetic energy
Electromagnetic field |
1 |
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1 B2 |
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energy density (in free |
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Energy density in |
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Energy flow (Poynting) |
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Mean flux density at a |
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ω4p2 sin2 θ |
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distance r from a short |
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(7.131) |
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32π2 0c3r3 r |
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oscillating dipole |
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Total mean power |
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ω4p2 |
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from oscillating |
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0 |
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W = 6π 0c3 |
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(7.132) |
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dipolea |
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Self-energy of a |
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1 |
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Utot = |
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φ(r)ρ(r) dτ |
(7.133) |
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charge distribution |
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volume |
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Energy of an assembly |
Utot = |
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i j |
Cij ViVj |
(7.134) |
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of capacitorsb |
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2 |
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Energy of an assembly |
Utot = |
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i j |
Lij IiIj |
(7.135) |
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of inductorsc |
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2 |
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Intrinsic dipole in an |
Udip = −p · E |
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(7.136) |
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electric field |
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Intrinsic dipole in a |
Udip = −m · B |
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(7.137) |
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magnetic field |
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Hamiltonian of a |
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|pm − qA|2 |
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charged particle in an |
H = |
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(7.138) |
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EM fieldd |
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2m |
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uenergy density
Eelectric field
Bmagnetic flux density
0 |
permittivity of free space |
µ0 |
permeability of free space |
Delectric displacement
Hmagnetic field strength
cspeed of light
Nenergy flow rate per unit area to the flow direction
p0 amplitude of dipole moment
rvector from dipole ( wavelength)
θangle between p and r
ωoscillation frequency
Wtotal mean radiated power
Utot |
total energy |
dτ |
volume element |
r |
position vector of dτ |
φelectrical potential
ρcharge density
Vi |
potential of ith capacitor |
Cij |
mutual capacitance between |
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capacitors i and j |
Lij |
mutual inductance between |
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inductors i and j |
Udip energy of dipole
pelectric dipole moment
mmagnetic dipole moment
HHamiltonian
pm particle momentum
qparticle charge
mparticle mass
Amagnetic vector potential
aSometimes called “Larmor’s formula.”
bCii is the self-capacitance of the ith body. Note that Cij = Cji. cLii is the self-inductance of the ith body. Note that Lij = Lji. dNewtonian limit, i.e., velocity c.