UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Supp HO # 6
Prof. Steven Errede
Supplemental Handout # 6
Symmetry Properties of Electromagnetism
The various field and source quantities, such as E , D, Ρ, H , B, Μ e.g. J e , J m , v , ε o , μo , L, S , . . .
etc. have various symmetry properties under symmetry operations such as:
P ≡ Parity (Space-Inversion, r → − r ) → Reflection in a mirror
T ≡ Time Reversal (e.g. particle motion, but run backwards in time)
C ≡ Electric Charge Conjugation (Charge of Particle → Charge of Antiparticle, e.g. e − → e + )
M = Magnetic Charge Conjugation (Magnetically Charged Particle → Magnetically charged
antiparticle, e.g. g N → g S )
The field and source quantities mentioned above fall into various generic mathematical classes of
objects, or quantities:
1) Scalar quantities under a given symmetry transformation, designated φ .
2) Pseudoscalar quantities under a given symmetry transformation, designated p.
3) Polar Vector quantities under a given symmetry transformation, designated V .
4) Axial, or Pseudo-Vector quantities under a given symmetry transformation, designated A .
5) Nth rank covariant / contravariant tensors under a given symmetry transformation,
γμ v
Tμ v , T μ v , TμV , Tσμ v , Tαβγ
,...
Note that, e.g.:
Electric charge q is ODD under electric charge conjugation: Ce − = e +
(i.e. q behaves as a pseudoscalar quantity p under C)
Magnetic charge g m is ODD under magnetic charge conjugation: Mg m− = g m+
(i.e. g m behaves as a pseudoscalar quantity p under M)
However note also that, e.g.:
Electric charge is EVEN under magnetic charge conjugation: Me− = e−
(i.e. q behaves as a scalar quantity φ under M.)
Magnetic charge is EVEN under electric charge conjugation: Cg m− = g m−
(i.e. g m behaves as a scalar quantity φ under C.)
Note also that (if ∃ no magnetic charges) the combined operations CPT = 1 (in any order)
(i.e. CPT = identity operator). If have magnetic charges, then CPTM =1 (in any order).
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
1
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Supp HO # 6
Prof. Steven Errede
In order to understand the distinction between Polar Vectors and Axial (or Pseudo)-Vectors
under a specific symmetry transformation, consider parity P (i.e. mirror-reflection) operation:
-component of V
unchanged under Parity
⊥ -component of V
changes sign under Parity
Polar Vector V
e.g. direction vector r
P, Parity is space-inversion
i.e.
r → −r
x → −x
i.e.
y → −y
z → −z
PV = V
PV⊥ = V⊥
-components of A
reversed under Parity
⊥ -components of A
unchanged under Parity
Axial (or Pseudo) Vector A
e.g. spinning top –
angular momentum L
i.e.
PA = − A
PA⊥ = A⊥
Time Reversal:
Particle
Moving with
Velocity v
Time-reversed
situation:
v
time
v′
time
v′ = Tv = −v (velocity is odd under Time reversal)
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©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Supp HO # 6
Prof. Steven Errede
Electric Current Flowing in a Long Wire:
∴
T vd = −vd
J efree = ne qvd
T J efree = − J efree
all
TI = −I
odd
TB = − B
under
TH = − H
T
Te − = e− (q is even under T)
I = J efree i Area
μo I × d
4π ∫ r 2
B = μH
B = μo H + Μ
B=
T μ = μ ( μ is even under T)
T Μ = −Μ
Current Flowing in a Loop:
∴
And
Tm = − m
T Μ = −Μ
(mag. dipole moment)
(magnetization)
By considering a parallel-plate capacitor, it can be seen that E , D and Ρ fields, p = electric
dipole moment, ε = permittivity, etc. are all even under time reversal.
TE = E
TΡ = Ρ
TD = D = ε E
(e.g. E =
1
q
rˆ )
4πε o r 2
Tq = + q (even under T)
Tε = ε
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
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UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Supp HO # 6
Prof. Steven Errede
Parity and Magnetic Fields
Electric Charge Conjugation & Magnetic Fields
Time Reversal & Magnetic Fields
Magnetic Charge Conjugation & Magnetic Fields
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©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Supp HO # 6
Prof. Steven Errede
Summary of Symmetry Properties of Kinematic & Electromagnetic Quantities
φ ≡
Scalar Quantity
p
≡ Pseudoscalar Quantity
V = Polar Vector
A = Axial Vector (Pseudo-Vector)
©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005 - 2008. All rights reserved.
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