# Maxwell`s equations ```Symbol
Meaning
Type of Object
~E
Electic Field
Vector field
~B
Magnetic Field
Vector field
e0
permitivity constant
constant
&micro;0
Permeability constant
constant
c
speed of light
constant
k
Coulomb’s constant:
σ
Closed orientable surface
C
Curve
simple closed piecewise smooth curve
~n
Outward unit normal vector, normal
to the surface σ
vector-valued function
~T
vector-valued function
qenc
Unit tangent vector, tangent to the
curve C
Charge enclosed by the surface
ρ
charge density
charge per unit volume
~J
current density vector
vector-valued function: magnitude
= current per unit area, direction =
same or opposite the velocity of the
moving charges, depending on their
sign
icondenc or ienc
conduction current enclosed by the
closed curve C
electric flux
ΦE
ΦB
RR
constant
magnetic flux
dS
σ
R
1
4πe0
ds
C
Surface integral; integrate with respect to the surface area of σ
Line integral; integrate with respect
to arc length of C; Note that integrals
H
of the form ~F • ~Tds can also be exHC
~ where r(t) is the
pressed as ~F • dr,
C
H
C
ds
S
~
dA
S
~
dA
C
~
ds
H
vector equation of C
Line integral over a closed curve
~
dA
Physicist’s symbol for a surface integral over a closed surface
Physicist’s symbol for a surface integral over a surface that is not closed.
Physicist’s symbol for a line integral
over a closed curve.
“area vector”
~
ds
“arc length vector”
R
H
vector-valued function; this is the
outward unit normal vector scaled
so that its magnitude accounts for
the area of the surface; it encompasses the ~ndS from the 3c formulas
involving surface integrals
vector-valued function; this is the
unit tangent vector scaled so that
its magnitude accounts for the arc
length; it encompasses the ~Tds or the
~ from the 3c formulas involving
dr
line integrals
Maxwell’s Equations
1. Gauss’ Law for Electricity
Integral Form
RR
~E • ~ndS =
ΦE =
|{z}
σ
3c
Differential Form
qenc
e0
5 • ~E = div(~E) =
ρ
e0
f lux
ΦE =
|{z}
4b
H
S
~ =
~E • dA
qenc
e0
f lux
2. Gauss’ law for magnetism
3c
4b
Integral Form
Differential Form
RR
~B • ~ndS = 0
ΦB =
5 • ~B = div(~B) = 0
σ
H
~ =0
ΦB = S ~B • dA
Ingegral Form
H
3c
~E • ~Tds =
C
I
H
C
~ = − dΦB
~E • dr
dt
~ = − dΦB = − d
~E • ds
dt
dt
| {z }
4b
Differential Form
C
R
S
~
5 &times; ~E = curl (~E) = − ∂∂tB
~
~B • dA
em f
4. Ampere/Maxwell’s Law
Ingegral Form
3c
4b
H
C
H
~B • ~Tds =
H
C
~ = &micro;0 e0 dΦE + &micro;0 ienc
~B • dr
dt
dΦE
~ ~
C B • ds = &micro;0 e0 dt + &micro;0 icondenc
H
R
d
~ ~
~ ~
C B • ds = &micro;0 e0 dt S E • dA + &micro;0 icondenc
Differential Form
5 &times; ~B
=
curl (~B)
=
&micro;0 ienc + e0 &micro;0
=
=
∂~E
∂t
4πk~
1 ∂~E
J+ 2
c2
c ∂t
~J
1 ∂~E
+ 2
e0 c 2
c ∂t
Some formulas were taken from the following website:
hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq.html
Thanks to Kaz Tarui and Katherine Meyer-Canales for their help in developing this
handout.
```