CHAPTER 6
Electromagnetic Wave Theory
6.1. Maxwell’s Equations
From all our discussions so far, we can summarize all of electrodynamics into five equations: (i) the
Lorentz force law, (ii) Gauss’ law for electrostatics and (iii) for magnetostatics, (iv) Faraday’s law, and
(v) Ampere’s law. These equations are written below:
⃗F = q ⃗E + ⃗v × B⃗
ρenc
∇ • ⃗E =
ε0
⃗
∇•B=0
∂B⃗
∇ × ⃗E = −
∂t
⃗
∇ × B = µ0 J ⃗
This was the state of electrodynamics before Maxwell. However, there is a fatal inconsistency in
Ampere’s law. We know that the divergence of a curl is always zero, for any vector. This means that
∇• ∇×B ⃗ =0
=⇒ ∇ • µ0⃗J = 0
=⇒ ∇ • ⃗J = 0
for all cases. We had also previously introduced the equation of continuity
∂ρ
+ ∇ • ⃗J = 0
∂t
So if ∇ • ⃗J is always zero, it implies that ∂ρ
∂t is also always zero. However this cannot always be true,
∂ρ
as ∂t = 0 implies a steady unchanging current. However, we know that unsteady currents also exist;
for example, AC currents change with time, and RC and LR circuits have exponentially changing
currents. So since ∂ρ ⃗
∂t = 0 is not universally true, it means that ∇ • J = 0 is also not universally true.
So Ampere’s law must be wrong or incomplete.
117
, CHAPTER 6. ELECTROMAGNETIC WAVE THEORY 118
The way Maxwell went about fixing the last equation was by investigating the continuity equation.
We have
∂ρ
∇ • ⃗J = −
∂t
∂
=− ε0 ∇ • ⃗E
∂t !
∂⃗E
= −∇ • ε0
∂t
!
∂⃗E
=⇒ ∇ • ⃗J + ε0 =0 (6.1.1)
∂t
which is true for all cases. Maxwell argued that if Ampere’s law was modified to be
!
⃗
⃗ = µ0
∇×B ⃗J + ε0 ∂E (6.1.2)
∂t
⃗
⃗ = µ0 ∇ • ⃗J + ε0 ∂E
then ∇ • ∇ × B ∂t which is zero for all possible cases (as per equation 6.1.1), and
thus the inconsistency was removed. He then defined the additional term he added to Ampere’s law
as the displacement current density.
⃗
⃗Jd = ε0 ∂E (6.1.3)
∂t
Thus, just as Faraday realized that a changing magnetic field gives rise to an electric field, Maxwell
also made the crucial discovery that a changing electric field gives rise to a magnetic field. The
reason that the displacement current density had not been detected by experiments was because it
was extremely small compared to the normal current density ⃗J that was being used in the laboratory.
Similarly we can also define the displacement current as
ˆ
∂⃗E ⃗ = ε0 ∂Φe
Id = ε0 • dS (6.1.4)
S ∂t ∂t
where Φe is the flux due to the electric field alone.
Finally, we can now write the integral form of the modified Ampere’s law as
˛
⃗ • d⃗l = µ0 (Ienc + Id ) = µ0 Ienc + µ0 ε0 ∂Φe
B (6.1.5)
L ∂t
Thus we can now summarize the correct Maxwell’s equations below. Their forms are different de-
pending upon what media we are investigating
Maxwell’s equations in free space
ρenc
∇ • ⃗E = ⃗ =0
∇•B
ε0
⃗
∂B ⃗
∇ × ⃗E = − ⃗ = µ0⃗J + µ0 ε0 ∂E
∇×B
∂t ∂t
Maxwell’s equations in free space
Electromagnetic Wave Theory
6.1. Maxwell’s Equations
From all our discussions so far, we can summarize all of electrodynamics into five equations: (i) the
Lorentz force law, (ii) Gauss’ law for electrostatics and (iii) for magnetostatics, (iv) Faraday’s law, and
(v) Ampere’s law. These equations are written below:
⃗F = q ⃗E + ⃗v × B⃗
ρenc
∇ • ⃗E =
ε0
⃗
∇•B=0
∂B⃗
∇ × ⃗E = −
∂t
⃗
∇ × B = µ0 J ⃗
This was the state of electrodynamics before Maxwell. However, there is a fatal inconsistency in
Ampere’s law. We know that the divergence of a curl is always zero, for any vector. This means that
∇• ∇×B ⃗ =0
=⇒ ∇ • µ0⃗J = 0
=⇒ ∇ • ⃗J = 0
for all cases. We had also previously introduced the equation of continuity
∂ρ
+ ∇ • ⃗J = 0
∂t
So if ∇ • ⃗J is always zero, it implies that ∂ρ
∂t is also always zero. However this cannot always be true,
∂ρ
as ∂t = 0 implies a steady unchanging current. However, we know that unsteady currents also exist;
for example, AC currents change with time, and RC and LR circuits have exponentially changing
currents. So since ∂ρ ⃗
∂t = 0 is not universally true, it means that ∇ • J = 0 is also not universally true.
So Ampere’s law must be wrong or incomplete.
117
, CHAPTER 6. ELECTROMAGNETIC WAVE THEORY 118
The way Maxwell went about fixing the last equation was by investigating the continuity equation.
We have
∂ρ
∇ • ⃗J = −
∂t
∂
=− ε0 ∇ • ⃗E
∂t !
∂⃗E
= −∇ • ε0
∂t
!
∂⃗E
=⇒ ∇ • ⃗J + ε0 =0 (6.1.1)
∂t
which is true for all cases. Maxwell argued that if Ampere’s law was modified to be
!
⃗
⃗ = µ0
∇×B ⃗J + ε0 ∂E (6.1.2)
∂t
⃗
⃗ = µ0 ∇ • ⃗J + ε0 ∂E
then ∇ • ∇ × B ∂t which is zero for all possible cases (as per equation 6.1.1), and
thus the inconsistency was removed. He then defined the additional term he added to Ampere’s law
as the displacement current density.
⃗
⃗Jd = ε0 ∂E (6.1.3)
∂t
Thus, just as Faraday realized that a changing magnetic field gives rise to an electric field, Maxwell
also made the crucial discovery that a changing electric field gives rise to a magnetic field. The
reason that the displacement current density had not been detected by experiments was because it
was extremely small compared to the normal current density ⃗J that was being used in the laboratory.
Similarly we can also define the displacement current as
ˆ
∂⃗E ⃗ = ε0 ∂Φe
Id = ε0 • dS (6.1.4)
S ∂t ∂t
where Φe is the flux due to the electric field alone.
Finally, we can now write the integral form of the modified Ampere’s law as
˛
⃗ • d⃗l = µ0 (Ienc + Id ) = µ0 Ienc + µ0 ε0 ∂Φe
B (6.1.5)
L ∂t
Thus we can now summarize the correct Maxwell’s equations below. Their forms are different de-
pending upon what media we are investigating
Maxwell’s equations in free space
ρenc
∇ • ⃗E = ⃗ =0
∇•B
ε0
⃗
∂B ⃗
∇ × ⃗E = − ⃗ = µ0⃗J + µ0 ε0 ∂E
∇×B
∂t ∂t
Maxwell’s equations in free space
