The_Cambridge_Handbook_of_Physics_Formulas

146 Electromagnetism
Electromagnetic energy
Electromagnetic field
energy density (in free u = 1
space)
2 ɛ 0E 2 + 1 B 2
(7.128)
2 µ 0
Energy density in
media
Energy flow (Poynting)
vector
u
E
B
energy density
electric field
magnetic flux density
ɛ 0 permittivity of free space
u = 1 2 (D ·E +B ·H) (7.129) µ 0 permeability of free space
D electric displacement
H magnetic field strength
N = E×H (7.130)
c
N
speed of light
energy flow rate per unit
area ⊥ to the flow direction
Mean flux density at a
distance r from a short 〈N〉 = ω4 p 2 0 sin2 r vector from dipole
θ
oscillating dipole
32π 2 ɛ 0 c 3 r 3 r (7.131) (≫wavelength)
θ angle between p and r
p 0 amplitude of dipole moment
ω oscillation frequency
Total mean power
from oscillating W = ω4 p 2 0 /2
dipole a 6πɛ 0 c 3 (7.132) W total mean radiated power
Self-energy of a
charge distribution
U tot = 1 2
Energy of an assembly
of capacitors b U tot = 1 2
Energy of an assembly
of inductors c U tot = 1 2
Intrinsic dipole in an
electric field
Intrinsic dipole in a
magnetic field
∫
volume
φ(r)ρ(r)dτ (7.133)
∑∑
C ij V i V j (7.134)
i
j
∑∑
i
j
U tot
dτ
r
φ
ρ
V i
C ij
total energy
volume element
position vector of dτ
electrical potential
charge density
potential of ith capacitor
mutual capacitance between
capacitors i and j
L ij I i I j (7.135) L ij mutual inductance between
inductors i and j
U dip = −p ·E (7.136)
U dip energy of dipole
p electric dipole moment
U dip = −m·B (7.137) m magnetic dipole moment
Hamiltonian of a
charged particle in an H = |p m −qA| 2
+qφ (7.138)
EM field d 2m
a Sometimes called “Larmor’s formula.”
b C ii is the self-capacitance of the ith body. Note that C ij = C ji .
c L ii is the self-inductance of the ith body. Note that L ij = L ji .
d Newtonian limit, i.e., velocity ≪ c.
H
p m
q
m
A
Hamiltonian
particle momentum
particle charge
particle mass
magnetic vector potential