This article summarizes equations in the theory of electromagnetism .
Definitions
Lorentz force on a charged particle (of charge q ) in motion (velocity v ), used as the definition of the E fieldB field Here subscripts e and m are used to differ between electric and magnetic charges . The definitions for monopoles are of theoretical interest, although real magnetic dipoles can be described using pole strengths. There are two possible units for monopole strength, Wb (Weber) and A m (Ampere metre). Dimensional analysis shows that magnetic charges relate by qm (Wb) = μ 0 qm (Am).
Initial quantities
Electric quantities
Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂ , d is the dipole moment between two point charges, the volume density of these is the polarization density P . Position vector r is a point to calculate the electric field; r′ is a point in the charged object. Contrary to the strong analogy between (classical) gravitation and electrostatics , there are no "centre of charge" or "centre of electrostatic attraction" analogues.
Electric transport
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Linear, surface, volumetric charge density
λe for Linear, σe for surface, ρe for volume.
q
e
=
∫
λ
e
d
ℓ
{\displaystyle q_{e}=\int \lambda _{e}\mathrm {d} \ell }
q
e
=
∬
σ
e
d
S
{\displaystyle q_{e}=\iint \sigma _{e}\mathrm {d} S}
q
e
=
∭
ρ
e
d
V
{\displaystyle q_{e}=\iiint \rho _{e}\mathrm {d} V}
C m−n , n = 1, 2, 3
[I][T][L]−n
Capacitance
C
C
=
d
q
/
d
V
{\displaystyle C=\mathrm {d} q/\mathrm {d} V\,\!}
V = voltage, not volume.
F = C V−1
[I]2 [T]4 [L]−2 [M]−1
Electric current
I
I
=
d
q
/
d
t
{\displaystyle I=\mathrm {d} q/\mathrm {d} t\,\!}
A
[I]
Electric current density
J
I
=
J
⋅
d
S
{\displaystyle I=\mathbf {J} \cdot \mathrm {d} \mathbf {S} }
A m−2
[I][L]−2
Displacement current density
J d
J
d
=
ϵ
0
(
∂
E
/
∂
t
)
=
∂
D
/
∂
t
{\displaystyle \mathbf {J} _{\mathrm {d} }=\epsilon _{0}\left(\partial \mathbf {E} /\partial t\right)=\partial \mathbf {D} /\partial t\,\!}
A m−2
[I][L]−2
Convection current density
J c
J
c
=
ρ
v
{\displaystyle \mathbf {J} _{\mathrm {c} }=\rho \mathbf {v} \,\!}
A m−2
[I][L]−2
Electric fields
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Electric field , field strength, flux density, potential gradient
E
E
=
F
/
q
{\displaystyle \mathbf {E} =\mathbf {F} /q\,\!}
N C−1 = V m−1
[M][L][T]−3 [I]−1
Electric flux
ΦE
Φ
E
=
∫
S
E
⋅
d
A
{\displaystyle \Phi _{E}=\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} \,\!}
N m2 C−1
[M][L]3 [T]−3 [I]−1
Absolute permittivity ;
ε
ϵ
=
ϵ
r
ϵ
0
{\displaystyle \epsilon =\epsilon _{r}\epsilon _{0}\,\!}
F m−1
[I]2 [T]4 [M]−1 [L]−3
Electric dipole moment
p
p
=
q
a
{\displaystyle \mathbf {p} =q\mathbf {a} \,\!}
a = charge separation
directed from -ve to +ve charge
C m
[I][T][L]
Electric Polarization, polarization density
P
P
=
d
⟨
p
⟩
/
d
V
{\displaystyle \mathbf {P} =\mathrm {d} \langle \mathbf {p} \rangle /\mathrm {d} V\,\!}
C m−2
[I][T][L]−2
Electric displacement field , flux density
D
D
=
ϵ
E
=
ϵ
0
E
+
P
{\displaystyle \mathbf {D} =\epsilon \mathbf {E} =\epsilon _{0}\mathbf {E} +\mathbf {P} \,}
C m−2
[I][T][L]−2
Electric displacement flux
ΦD
Φ
D
=
∫
S
D
⋅
d
A
{\displaystyle \Phi _{D}=\int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} \,\!}
C
[I][T]
Absolute electric potential , EM scalar potential relative to point
r
0
{\displaystyle r_{0}\,\!}
Theoretical:
r
0
=
∞
{\displaystyle r_{0}=\infty \,\!}
r
0
=
R
e
a
r
t
h
{\displaystyle r_{0}=R_{\mathrm {earth} }\,\!}
φ ,V
V
=
−
W
∞
r
q
=
−
1
q
∫
∞
r
F
⋅
d
r
=
−
∫
r
1
r
2
E
⋅
d
r
{\displaystyle V=-{\frac {W_{\infty r)){q))=-{\frac {1}{q))\int _{\infty }^{r}\mathbf {F} \cdot \mathrm {d} \mathbf {r} =-\int _{r_{1))^{r_{2))\mathbf {E} \cdot \mathrm {d} \mathbf {r} \,\!}
V = J C−1
[M] [L]2 [T]−3 [I]−1
Voltage , Electric potential difference
Δφ ,ΔV
Δ
V
=
−
Δ
W
q
=
−
1
q
∫
r
1
r
2
F
⋅
d
r
=
−
∫
r
1
r
2
E
⋅
d
r
{\displaystyle \Delta V=-{\frac {\Delta W}{q))=-{\frac {1}{q))\int _{r_{1))^{r_{2))\mathbf {F} \cdot \mathrm {d} \mathbf {r} =-\int _{r_{1))^{r_{2))\mathbf {E} \cdot \mathrm {d} \mathbf {r} \,\!}
V = J C−1
[M] [L]2 [T]−3 [I]−1
Magnetic quantities
Magnetic transport
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Linear, surface, volumetric pole density
λm for Linear, σm for surface, ρm for volume.
q
m
=
∫
λ
m
d
ℓ
{\displaystyle q_{m}=\int \lambda _{m}\mathrm {d} \ell }
q
m
=
∬
σ
m
d
S
{\displaystyle q_{m}=\iint \sigma _{m}\mathrm {d} S}
q
m
=
∭
ρ
m
d
V
{\displaystyle q_{m}=\iiint \rho _{m}\mathrm {d} V}
Wb m−n A m(−n + 1) , n = 1, 2, 3
[L]2 [M][T]−2 [I]−1 (Wb)[I][L] (Am)
Monopole current
Im
I
m
=
d
q
m
/
d
t
{\displaystyle I_{m}=\mathrm {d} q_{m}/\mathrm {d} t\,\!}
Wb s−1 A m s−1
[L]2 [M][T]−3 [I]−1 (Wb)[I][L][T]−1 (Am)
Monopole current density
J m
I
=
∬
J
m
⋅
d
A
{\displaystyle I=\iint \mathbf {J} _{\mathrm {m} }\cdot \mathrm {d} \mathbf {A} }
Wb s−1 m−2 A m−1 s−1
[M][T]−3 [I]−1 (Wb)[I][L]−1 [T]−1 (Am)
Magnetic fields
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Magnetic field , field strength, flux density, induction field
B
F
=
q
e
(
v
×
B
)
{\displaystyle \mathbf {F} =q_{e}\left(\mathbf {v} \times \mathbf {B} \right)\,\!}
T = N A−1 m−1 = Wb m−2
[M][T]−2 [I]−1
Magnetic potential , EM vector potential
A
B
=
∇
×
A
{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }
T m = N A−1 = Wb m3
[M][L][T]−2 [I]−1
Magnetic flux
ΦB
Φ
B
=
∫
S
B
⋅
d
A
{\displaystyle \Phi _{B}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} \,\!}
Wb = T m2
[L]2 [M][T]−2 [I]−1
Magnetic permeability
μ
{\displaystyle \mu \,\!}
μ
=
μ
r
μ
0
{\displaystyle \mu \ =\mu _{r}\,\mu _{0}\,\!}
V·s·A−1 ·m−1 = N·A−2 = T·m·A−1 = Wb·A−1 ·m−1
[M][L][T]−2 [I]−2
Magnetic moment , magnetic dipole moment
m , μB , Π
Two definitions are possible:
using pole strengths,
m
=
q
m
a
{\displaystyle \mathbf {m} =q_{m}\mathbf {a} \,\!}
using currents:
m
=
N
I
A
n
^
{\displaystyle \mathbf {m} =NIA\mathbf {\hat {n)) \,\!}
a = pole separation
N is the number of turns of conductor
A m2
[I][L]2
Magnetization
M
M
=
d
⟨
m
⟩
/
d
V
{\displaystyle \mathbf {M} =\mathrm {d} \langle \mathbf {m} \rangle /\mathrm {d} V\,\!}
A m−1
[I] [L]−1
Magnetic field intensity, (AKA field strength)
H
Two definitions are possible:
most common:
B
=
μ
H
=
μ
0
(
H
+
M
)
{\displaystyle \mathbf {B} =\mu \mathbf {H} =\mu _{0}\left(\mathbf {H} +\mathbf {M} \right)\,}
using pole strengths,[1]
H
=
F
/
q
m
{\displaystyle \mathbf {H} =\mathbf {F} /q_{m}\,}
A m−1
[I] [L]−1
Intensity of magnetization , magnetic polarization
I , J
I
=
μ
0
M
{\displaystyle \mathbf {I} =\mu _{0}\mathbf {M} \,\!}
T = N A−1 m−1 = Wb m−2
[M][T]−2 [I]−1
Self Inductance
L
Two equivalent definitions are possible:
L
=
N
(
d
Φ
/
d
I
)
{\displaystyle L=N\left(\mathrm {d} \Phi /\mathrm {d} I\right)\,\!}
L
(
d
I
/
d
t
)
=
−
N
V
{\displaystyle L\left(\mathrm {d} I/\mathrm {d} t\right)=-NV\,\!}
H = Wb A−1
[L]2 [M] [T]−2 [I]−2
Mutual inductance
M
Again two equivalent definitions are possible:
M
1
=
N
(
d
Φ
2
/
d
I
1
)
{\displaystyle M_{1}=N\left(\mathrm {d} \Phi _{2}/\mathrm {d} I_{1}\right)\,\!}
M
(
d
I
2
/
d
t
)
=
−
N
V
1
{\displaystyle M\left(\mathrm {d} I_{2}/\mathrm {d} t\right)=-NV_{1}\,\!}
1,2 subscripts refer to two conductors/inductors mutually inducing voltage/ linking magnetic flux through each other. They can be interchanged for the required conductor/inductor;
M
2
=
N
(
d
Φ
1
/
d
I
2
)
{\displaystyle M_{2}=N\left(\mathrm {d} \Phi _{1}/\mathrm {d} I_{2}\right)\,\!}
M
(
d
I
1
/
d
t
)
=
−
N
V
2
{\displaystyle M\left(\mathrm {d} I_{1}/\mathrm {d} t\right)=-NV_{2}\,\!}
H = Wb A−1
[L]2 [M] [T]−2 [I]−2
Gyromagnetic ratio (for charged particles in a magnetic field)
γ
ω
=
γ
B
{\displaystyle \omega =\gamma B\,\!}
Hz T−1
[M]−1 [T][I]
Electric circuits
DC circuits, general definitions
AC circuits
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Resistive load voltage
VR
V
R
=
I
R
R
{\displaystyle V_{R}=I_{R}R\,\!}
V = J C−1
[M] [L]2 [T]−3 [I]−1
Capacitive load voltage
VC
V
C
=
I
C
X
C
{\displaystyle V_{C}=I_{C}X_{C}\,\!}
V = J C−1
[M] [L]2 [T]−3 [I]−1
Inductive load voltage
VL
V
L
=
I
L
X
L
{\displaystyle V_{L}=I_{L}X_{L}\,\!}
V = J C−1
[M] [L]2 [T]−3 [I]−1
Capacitive reactance
XC
X
C
=
1
ω
d
C
{\displaystyle X_{C}={\frac {1}{\omega _{\mathrm {d} }C))\,\!}
Ω−1 m−1
[I]2 [T]3 [M]−2 [L]−2
Inductive reactance
XL
X
L
=
ω
d
L
{\displaystyle X_{L}=\omega _{d}L\,\!}
Ω−1 m−1
[I]2 [T]3 [M]−2 [L]−2
AC electrical impedance
Z
V
=
I
Z
{\displaystyle V=IZ\,\!}
Z
=
R
2
+
(
X
L
−
X
C
)
2
{\displaystyle Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2))}\,\!}
Ω−1 m−1
[I]2 [T]3 [M]−2 [L]−2
Phase constant
δ, φ
tan
ϕ
=
X
L
−
X
C
R
{\displaystyle \tan \phi ={\frac {X_{L}-X_{C)){R))\,\!}
dimensionless
dimensionless
AC peak current
I 0
I
0
=
I
r
m
s
2
{\displaystyle I_{0}=I_{\mathrm {rms} }{\sqrt {2))\,\!}
A
[I]
AC root mean square current
I rms
I
r
m
s
=
1
T
∫
0
T
[
I
(
t
)
]
2
d
t
{\displaystyle I_{\mathrm {rms} }={\sqrt ((\frac {1}{T))\int _{0}^{T}\left[I\left(t\right)\right]^{2}\mathrm {d} t))\,\!}
A
[I]
AC peak voltage
V 0
V
0
=
V
r
m
s
2
{\displaystyle V_{0}=V_{\mathrm {rms} }{\sqrt {2))\,\!}
V = J C−1
[M] [L]2 [T]−3 [I]−1
AC root mean square voltage
V rms
V
r
m
s
=
1
T
∫
0
T
[
V
(
t
)
]
2
d
t
{\displaystyle V_{\mathrm {rms} }={\sqrt ((\frac {1}{T))\int _{0}^{T}\left[V\left(t\right)\right]^{2}\mathrm {d} t))\,\!}
V = J C−1
[M] [L]2 [T]−3 [I]−1
AC emf, root mean square
E
r
m
s
,
⟨
E
⟩
{\displaystyle {\mathcal {E))_{\mathrm {rms} },{\sqrt {\langle {\mathcal {E))\rangle ))\,\!}
E
r
m
s
=
E
m
/
2
{\displaystyle {\mathcal {E))_{\mathrm {rms} }={\mathcal {E))_{\mathrm {m} }/{\sqrt {2))\,\!}
V = J C−1
[M] [L]2 [T]−3 [I]−1
AC average power
⟨
P
⟩
{\displaystyle \langle P\rangle \,\!}
⟨
P
⟩
=
E
I
r
m
s
cos
ϕ
{\displaystyle \langle P\rangle ={\mathcal {E))I_{\mathrm {rms} }\cos \phi \,\!}
W = J s−1
[M] [L]2 [T]−3
Capacitive time constant
τC
τ
C
=
R
C
{\displaystyle \tau _{C}=RC\,\!}
s
[T]
Inductive time constant
τL
τ
L
=
L
/
R
{\displaystyle \tau _{L}=L/R\,\!}
s
[T]
Magnetic circuits
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Magnetomotive force , mmf
F ,
F
,
M
{\displaystyle {\mathcal {F)),{\mathcal {M))}
M
=
N
I
{\displaystyle {\mathcal {M))=NI}
N = number of turns of conductor
A
[I]
Electromagnetism
Electric fields
General Classical Equations
Magnetic fields and moments
General classical equations
Electric circuits and electronics
Below N = number of conductors or circuit components. Subscript net refers to the equivalent and resultant property value.
Physical situation
Nomenclature
Series
Parallel
Resistors and conductors
Ri = resistance of resistor or conductor i Gi = conductance of resistor or conductor i
R
n
e
t
=
∑
i
=
1
N
R
i
{\displaystyle R_{\mathrm {net} }=\sum _{i=1}^{N}R_{i}\,\!}
1
G
n
e
t
=
∑
i
=
1
N
1
G
i
{\displaystyle {1 \over G_{\mathrm {net} ))=\sum _{i=1}^{N}{1 \over G_{i))\,\!}
1
R
n
e
t
=
∑
i
=
1
N
1
R
i
{\displaystyle {1 \over R_{\mathrm {net} ))=\sum _{i=1}^{N}{1 \over R_{i))\,\!}
G
n
e
t
=
∑
i
=
1
N
G
i
{\displaystyle G_{\mathrm {net} }=\sum _{i=1}^{N}G_{i}\,\!}
Charge, capacitors, currents
Ci = capacitance of capacitor i qi = charge of charge carrier i
q
n
e
t
=
∑
i
=
1
N
q
i
{\displaystyle q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!}
1
C
n
e
t
=
∑
i
=
1
N
1
C
i
{\displaystyle {1 \over C_{\mathrm {net} ))=\sum _{i=1}^{N}{1 \over C_{i))\,\!}
I
n
e
t
=
I
i
{\displaystyle I_{\mathrm {net} }=I_{i}\,\!}
q
n
e
t
=
∑
i
=
1
N
q
i
{\displaystyle q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!}
C
n
e
t
=
∑
i
=
1
N
C
i
{\displaystyle C_{\mathrm {net} }=\sum _{i=1}^{N}C_{i}\,\!}
I
n
e
t
=
∑
i
=
1
N
I
i
{\displaystyle I_{\mathrm {net} }=\sum _{i=1}^{N}I_{i}\,\!}
Inductors
Li = self-inductance of inductor i Lij = self-inductance element ij of L matrixMij = mutual inductance between inductors i and j
L
n
e
t
=
∑
i
=
1
N
L
i
{\displaystyle L_{\mathrm {net} }=\sum _{i=1}^{N}L_{i}\,\!}
1
L
n
e
t
=
∑
i
=
1
N
1
L
i
{\displaystyle {1 \over L_{\mathrm {net} ))=\sum _{i=1}^{N}{1 \over L_{i))\,\!}
V
i
=
∑
j
=
1
N
L
i
j
d
I
j
d
t
{\displaystyle V_{i}=\sum _{j=1}^{N}L_{ij}{\frac {\mathrm {d} I_{j)){\mathrm {d} t))\,\!}
Circuit
DC Circuit equations
AC Circuit equations
Series circuit equations
RC circuits
Circuit equation
R
d
q
d
t
+
q
C
=
E
{\displaystyle R{\frac {\mathrm {d} q}{\mathrm {d} t))+{\frac {q}{C))={\mathcal {E))\,\!}
Capacitor charge
q
=
C
E
(
1
−
e
−
t
/
R
C
)
{\displaystyle q=C{\mathcal {E))\left(1-e^{-t/RC}\right)\,\!}
Capacitor discharge
q
=
C
E
e
−
t
/
R
C
{\displaystyle q=C{\mathcal {E))e^{-t/RC}\,\!}
RL circuits
Circuit equation
L
d
I
d
t
+
R
I
=
E
{\displaystyle L{\frac {\mathrm {d} I}{\mathrm {d} t))+RI={\mathcal {E))\,\!}
Inductor current rise
I
=
E
R
(
1
−
e
−
R
t
/
L
)
{\displaystyle I={\frac {\mathcal {E)){R))\left(1-e^{-Rt/L}\right)\,\!}
Inductor current fall
I
=
E
R
e
−
t
/
τ
L
=
I
0
e
−
R
t
/
L
{\displaystyle I={\frac {\mathcal {E)){R))e^{-t/\tau _{L))=I_{0}e^{-Rt/L}\,\!}
LC circuits
Circuit equation
L
d
2
q
d
t
2
+
q
/
C
=
E
{\displaystyle L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2))}+q/C={\mathcal {E))\,\!}
Circuit equation
L
d
2
q
d
t
2
+
q
/
C
=
E
sin
(
ω
0
t
+
ϕ
)
{\displaystyle L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2))}+q/C={\mathcal {E))\sin \left(\omega _{0}t+\phi \right)\,\!}
Circuit resonant frequency
ω
r
e
s
=
1
/
L
C
{\displaystyle \omega _{\mathrm {res} }=1/{\sqrt {LC))\,\!}
Circuit charge
q
=
q
0
cos
(
ω
t
+
ϕ
)
{\displaystyle q=q_{0}\cos(\omega t+\phi )\,\!}
Circuit current
I
=
−
ω
q
0
sin
(
ω
t
+
ϕ
)
{\displaystyle I=-\omega q_{0}\sin(\omega t+\phi )\,\!}
Circuit electrical potential energy
U
E
=
q
2
/
2
C
=
Q
2
cos
2
(
ω
t
+
ϕ
)
/
2
C
{\displaystyle U_{E}=q^{2}/2C=Q^{2}\cos ^{2}(\omega t+\phi )/2C\,\!}
Circuit magnetic potential energy
U
B
=
Q
2
sin
2
(
ω
t
+
ϕ
)
/
2
C
{\displaystyle U_{B}=Q^{2}\sin ^{2}(\omega t+\phi )/2C\,\!}
RLC Circuits
Circuit equation
L
d
2
q
d
t
2
+
R
d
q
d
t
+
q
C
=
E
{\displaystyle L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2))}+R{\frac {\mathrm {d} q}{\mathrm {d} t))+{\frac {q}{C))={\mathcal {E))\,\!}
Circuit equation
L
d
2
q
d
t
2
+
R
d
q
d
t
+
q
C
=
E
sin
(
ω
0
t
+
ϕ
)
{\displaystyle L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2))}+R{\frac {\mathrm {d} q}{\mathrm {d} t))+{\frac {q}{C))={\mathcal {E))\sin \left(\omega _{0}t+\phi \right)\,\!}
Circuit charge
q
=
q
0
e
T
−
R
t
/
2
L
cos
(
ω
′
t
+
ϕ
)
{\displaystyle q=q_{0}eT^{-Rt/2L}\cos(\omega 't+\phi )\,\!}
![{\displaystyle I_{\mathrm {rms} }={\sqrt ((\frac {1}{T))\int _{0}^{T}\left[I\left(t\right)\right]^{2}\mathrm {d} t))\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1def0a7f6e77cb4c21b955601a657047a645a97)
![{\displaystyle V_{\mathrm {rms} }={\sqrt ((\frac {1}{T))\int _{0}^{T}\left[V\left(t\right)\right]^{2}\mathrm {d} t))\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a47cb2bc8518d427829d5d187367748f99a3dcf)
