Maxwell's Equation
OVERIEW
Basics
Maxwell's Equation
Gauss's Law
Gauss's Law of Magnetism
Faraday's Law
Ampere-Maxwell Law
Electromagnetic Wave Equation
Reference
Basics
Electric Field
Electric field is the physical field surrounding an electric charge defined as
the force per unit charge. We know from Coulomb's Law that the force
exerted by a charge q1 on another charge q0 at a distance x and x0 ,
respectively, from origin is given by,
F =
1
q1 q0
x
^ 1,0
4πε0 (x1 − x0 )2
Then, the electric field at the same location created by charge q1 is given
by,
E(r0 ) =
F
1
q1
^ 1,0
=
x
q0
4πε0 (x1 − x0 )2
Motivation to define this new vector function as electric field
 It makes it easy to understand the effect that a given set of charges
produces on another set. This problem can be conveniently divided
into two parts by introducing the electrostatic field, for then we can
(a) calculate the field due to a given distribution of charges without
worrying about the effect these charges have on other charges in
Maxwell's Equation
1
the vicinity and (b) calculate the effect a given field has on charges
placed in it without worrying about the distribution of charges that
produced the field.
 It turns out that all classical electromagnetic theory can be codified
in terms of four equations, called Maxwell's equations, which relate
fields (electric and magnetic) to each other and to the charges and
currents which produce them.
Thus, electromagnetism is a field theory and the electric field ultimately
plays a role and assumes an importance which far transcends its simple
elementary definition as "force per unit charge."
Very often it is convenient to treat a distribution of electric charge as if it
were continuous. To do this, we proceed as follows.
Suppose in some region of space of volume a V the total electric charge is
a Q. We define the average charge density in ΔV as,
ρˉΔV =
ΔQ
ΔV
Using this, we can define the charge density at the point (x, y, z), denoted
ρ(x, y, z), by taking the limit of ρˉΔV as ΔV shrinks down about the point
(x, y, z):
⁍
Thus, the electric charge in some region of volume V can then be
expressed as the triple integral of ρ(x, y, z) over the volume V ; that is,
Q = ∭ ρ(x, y, z) dV
V
Then, the Electric Field at distance x contributed by a continuous
distribution of charges at point x′ , is given by,
1
ρ(x′ ) dV ′
E(r0 ) =
∭
x
^
′
2
4πε0
V (x − x)
′
where x
^ is the unit vector pointing from x′ to x.
Surface Integral and Divergence
Maxwell's Equation
2
In three-dimensional Cartesian coordinates, the divergence of a
continuously differentiable
vector field F
= Fx i + Fy j + Fz k is defined as the scalar valued function
obtained by dot product,
div F = ∇ ⋅ F = (
∂ ∂ ∂
∂Fx
∂Fy
∂Fz
, , ) ⋅ (Fx , Fy , Fz ) =
+
+
∂x ∂y ∂z
∂x
∂y
∂z
Line Integral and Curl
The curl of a vector field F
= Fx i + Fy j + Fz k, denoted by ∇ × F is
given by the cross product.
∇×F=
∇×F=(
∣ ^
^
^ ∣
k
∂
∂x
∂
∂y
∂
∂z
∣ Fx
Fy
Fz ∣
∂Fz
∂Fy
∂Fx
∂Fz
∂Fy
∂Fx ^
−
) ^ + (
−
) ^ + (
−
)k
∂y
∂z
∂z
∂x
∂x
∂y
Gradient
💡
There is a single EM field that changes its appearance from
for different observers.
E to B
Maxwell's Equation
Introduction
Maxwell's equations are a set of coupled partial differential equations that,
together with the Lorentz force law, form the foundation of classical
electromagnetism, classical optics, and electric circuits.
All Integral equations as GIF
∬ E ⋅n
^ dA =
S
Maxwell's Equation
q
ε0
3
∬ B ⋅n
^ dA = 0
S
∮ E ⋅ dl = −
Maxwell's Equation
d
∬ B⋅n
^ dA
dt S
4
∮ B ⋅ dl = μ0[I + ε0
d
∬ E ⋅n
^ dA]
dt S
Maxwell's equations consists of 4 equations describing 4 laws that can be
expressed in two forms.
Integral Form
∬ E ⋅n
^ dA =
S
∑ Qenclosed
ε0
∬ B ⋅n
^ dA = 0
S
∮ E ⋅ dl = − ∬
Maxwell's Equation
S
∂B
⋅n
^ dA
∂t
5
∮ B ⋅ dl = μ0 [ ∑ Ienclosed + ε0 ∬
S
∂E
⋅n
^ dA]
∂t
All Equations
Differential Form
∇⋅E=
ρ
ε0
∇⋅B = 0
∇×E=−
∂B
∂t
∇ × B = μ0 (J + ε0
∂E
)
∂t
Gauss's Law
Integral form
Maxwell's Equation
6
∬ E ⋅n
^ dA =
S
∑ Qenclosed
ε0
Law
Gauss's law, also known as Gauss's flux theorem, is a law relating the
distribution of electric charge to the resulting electric field.
It states that the net electric flux of the electric field through any
arbitrary closed surface is equal to
within that closed surface,
1
times the net electric charge
ε0
ΦE =
Q
ε0
where ΦE is the electric flux through a closed surface S enclosing any
volume V , Q is the total charge enclosed within V , and ϵ0 is
the electric constant.
The electric flux ΦE is defined as a surface integral of the electric field:
ΦE = ∬ E ⋅ dA
S
where E is the electric field, dA is a vector representing
an infinitesimal element of area of the surface, and ⋅ represents the dot
product of two vectors.
The law alone is insufficient to determine the electric field across a
surface enclosing any charge distribution because the total flux
through that given surface gives little information about the electric
field, and can go in and out of the surface in arbitrarily complicated
patterns.
An exception is if there is some symmetry in the problem, which
mandates that the electric field passes through the surface in a uniform
way. Then, if the total flux is known, the field itself can be deduced at
every point. Common examples of symmetries which lend themselves
to Gauss's law include: cylindrical symmetry, planar symmetry, and
spherical symmetry.
Where no such symmetry exists, Gauss's law can be used in its
differential form, which states that the divergence of the electric field is
Maxwell's Equation
7
proportional to the local density of charge.
Explanation
Maxwell's Equation
8
Differential form
∇⋅E=
ρ
ε0
Law
The electric field produced by electric charge diverges from positive
charge and converges upon negative charge.
Explanation
Derive from Integral form using small volume box ΔV .
Maxwell's Equation
9
Gauss's Law of Magnetism
Integral form
∬ B ⋅n
^ dA = 0
S
Law
The left-hand side of this equation is called the net flux of the
magnetic field B out of the surface S , and Gauss's law for
magnetism states that it is always zero.
The law in this form states that for each volume element in space, there
are exactly the same number of "magnetic field lines" entering and
exiting the volume. No total "magnetic charge" can build up in any point
in space.
There are no magnetic flow sources, and the magnetic flux lines
always close upon themselves.
Also called the law of conservation of magnetic flux
Maxwell's Equation
10
For example, the south pole of the magnet is exactly as strong as the
north pole, and free-floating south poles without accompanying north
poles (magnetic monopoles) doesn't exist. In contrast, this is not true
for other fields such as electric fields or gravitational fields, where total
electric charge or mass can build up in a volume of space.
The law is also called "Absence of free magnetic poles". The
assumption that there are no magnetic monopoles.
Explanation
Differential form
∇⋅B = 0
Law
The divergence of the magnetic field at any point is zero.
The assumption that there are no magnetic monopoles.
Explanation
Maxwell's Equation
11
Faraday's Law
Integral form
∮ E ⋅ dl = − ∬
S
∂B
⋅n
^ dA
∂t
Law
A changing magnetic field induces a circulating electric field.
Changing magnetic flux through a surface induces an electromotive
force EMF in any boundary path of that surface.
💡
The way I see this is, whenever there is a changing
magnetic field, a circulating electric field always appears,
perpendicularly. And if we place a wire let's say in this
field, the electrons experience force giving rise to EMF
induction.
The voltage accumulated around a closed circuit is proportional to
the time rate of change of the magnetic flux it encloses.
Explanation
Maxwell's Equation
12
Differential form
∇×E=−
∂B
∂t
Law
A circulating electric field is produced by a magnetic field that changes
with time.
The Maxwell–Faraday equation states that a time-varying magnetic
field always accompanies a spatially varying (also possibly timevarying), non-conservative electric field, and vice versa.
Explanation
Ampere-Maxwell Law
Integral form
Maxwell's Equation
13
∮ B ⋅ dl = μ0 [ ∑ Ienclosed + ε0 ∬
S
∂E
⋅n
^ dA]
∂t
Law
An electric current I or a changing electric flux through a surface
produces a circulating magnetic field around any path that bounds
that surface.
Electric currents and changes in electric fields are proportional to
the magnetic fields circulating about the areas where they
accumulate.
Explanation
The second term was added by Maxwell which explains the case of a
capacitor. Here the surface bounded by closed path C is bulged such
that the current ic doesn't penetrate this surface, meaning it is not
enclosed in this surface. However, since the electric field penetrate this
surface, which appears in the equation above that gives the circulating
magnetic field around it.
Maxwell's Equation
14
Differential form
∇ × B = μ0 (J + ε0
∂E
)
∂t
Law
A circulating magnetic field is produced by an electric current and by an
electric field that changes with time.
Explanation
Derive from integral form using closed path with small area ΔA.
Maxwell's Equation
15
Maxwell's Equation
16
Electromagnetic Wave Equation
Derivation
To obtain the electromagnetic wave equation in a vacuum using the
modern method, we begin with the modern 'Heaviside' form of Maxwell's
equations. In a vacuum- and charge-free space, these equations are:
∇⋅E= 0
∇×E=−
∇⋅B = 0
∂B
∂t
∇ × B = μ0 ε0
∂E
∂t
These are the general Maxwell's equations specialized to the case with
charge and current both set to zero. Taking the curl of the curl equations
gives:
∂B
∂
∂2 E
∇ × (∇ × E) = ∇ × (−
) = − (∇ × B) = −μ0 ε0 2
∂t
∂t
∂t
∂E
∂
∂2 B
∇ × (∇ × B) = ∇ × (μ0 ε0
) = μ0 ε0 (∇ × E) = −μ0 ε0 2
∂t
∂t
∂t
We can use the vector identity
Maxwell's Equation
17
∇ × (∇ × V) = ∇ (∇ ⋅ V) − ∇2 V
where V is any vector function of space.
Since
∇⋅E= 0
∇⋅B = 0
then the first term on the right in the identity vanishes and we obtain the
wave equations:
∂2 E
μ0 ε0 2 = ∇2 E
∂t
∂2 E
∂2 E ∂2 E ∂2 E
μ0 ε0 2 = ( 2 +
+
)
∂t
∂x
∂y2
∂z 2
similarly,
μ0 ε0
∂2 B
∂2 B
∂2 B
∂2 B
=
(
+
+
)
∂t2
∂x2
∂y2
∂z 2
which is a wave equation, where, the speed of the wave c0 is given by,
c0 =
1
= 2.99792458 × 108 m/s
μ0 ε0
is the speed of light in free space.
EM Wave spectrum
Maxwell's Equation
18
Reference
Plain Explanation of Maxwell Equation
Lecture 26 Maxwell Equation by Prof. Carlson
Accelerated Charges Radiating Electromagnetic Waves
Episode 39 Maxwell's Equation - The Mechanical Universe by Caltech
Electromagnetic Waves - with Sir Lawrence Bragg
Electric and Magnetic Field from Power Lines by BC Hydro
Moving magnet and conductor problem
The moving magnet and conductor problem is a famous thought
experiment, originating in the 19th century, concerning the intersection of
classical electromagnetism and special relativity.
In it, the current in a conductor moving with constant velocity, v, with
respect to a magnet is calculated in the frame of reference of the magnet
and in the frame of reference of the conductor. The observable quantity in
the experiment, the current, is the same in either case, in accordance with
the basic principle of relativity, which states: "Only relative motion is
observable; there is no absolute standard of rest".
If one is in a frame that is at rest with respect to the magnet and the
conductor is moving, then the induced emf is a result of the Lorentz force
acting on the charge carries moving in a magnetic field.
Maxwell's Equation
19
On the other hand, if one is in a frame at rest with respect to the conductor
and the magnet is moving, the induced emf results from the variation in the
magnetic flux which produces electric field.
Thus, according to Maxwell's equations, the charges in the conductor
experience a magnetic force in the frame of the magnet and an electric
force in the frame of the conductor. The same phenomenon would seem to
have two different descriptions depending on the frame of reference of the
observer.
Which, in Einstein's word, "For if the magnet is in motion and the conductor
at rest, there arises in the neighbourhood of the magnet an electric field
with a certain definite energy, producing a current at the places where
parts of the conductor are situated. But if the magnet is stationary and the
conductor in motion, no electric field arises in the neighbourhood of the
magnet. In the conductor, however, we find an electromotive force, to
which in itself there is no corresponding energy, but which gives rise –
assuming equality of relative motion in the two cases discussed – to
electric currents of the same path and intensity as those produced by the
electric forces in the former case."
Maxwell's Equation
20