Download 3 Maxwell`s equations and material equations

3
Maxwell’s equations and material
equations
Contents
3.1
Macroscopic Maxwell’s equations
3.2
Material equations and LIH
3.3
The wave equation
3.4
Vector and scalar potentials
Keywords: Maxwell equations, material equations, LIH, vector potential,
scalar potential, wave equation and electromagnetic waves
Ref: J. D. Jackson: Classical Electrodynamics; A. Sommerfeld: Electrodynamics.
3.1 Macroscopic Maxwell’s equations
Any electromagnetic field in certain medium respects the following four equations in SI units,
∇ · D = ρ,
∇×E=−
(3.1)
∂B
,
∂t
∇ · B = 0,
∇×H=j+
(3.2)
(3.3)
∂D
,
∂t
(3.4)
2
3 Maxwell’s equations and material equations
where E and B are the electric and the magnetic fields respectively. The
above equations are called macroscopic Maxwell’s equations. The first and
the third equations are independently known as Gauss’ law for electrostatics
and magnetostatics respectively. When we write them in Maxwell’s equation
the status of these are raised from statics to dynamics. In other words the laws
are valid even if the charges are moving. The second equation is nothing but
the Faraday’s law whereas the fourth equation, the Ampere’s law modified
by Maxwell with an additional term Ḋ, is called the Ampere-Maxwell law. It
says that a by time varying magnetic field one can generate an electric field
and it complements the Faraday’s law. These equations look slightly different
in CGS and other units. The displacement vector D roughly tells how the
medium is distorted when an electric field is applied or in other words it is
a measure of response of the medium to the field. H is the analogue of D
for the magnetic field. ρ represents the charge density and j is the current
density i.e. current per unit area. These two densities are often termed as
sources to the field. The ρ is measured in coulomb/m3 and j is expressed in
ampere/m2 in SI units. The electric field is measured in newton/coulomb
and the magnetic field SI unit is tesla. Permanent magnets are outside the
scope of Maxwell’s electrodynamics which is essentially a classical description
whereas the the permanent magnets necessarily need a quantum description.
NPTEL Course: Wave Propagation in Continuous Media
3.2 Material equations and LIH
3
3.2 Material equations and LIH
If one is supposed to find the electromagnetic field for a given set of ρ and j
one has to solve the Maxwell’s equations. But the problem is that there are
twelve unknowns since there are four vectors, E, D, B and H, each having
three components and there are only eight equations(each vector equation
is equivalent to three scalar equations). So to solve them we have to make
some assumptions. These assumptions give us the material equations. Like
Hooke’s law if the fields are not very strong one assumes D and H are linearly
proportional to E and B. The E and B are analogues of the external or
applied force (proportional to stress) and D and H are analogues of the
strain in the medium.
D = ǫE,
(3.5)
1
B,
µ
(3.6)
j = σE.
(3.7)
H=
Such materials where these relations are valid are known as Linear-IsotropicHomogeneous or LIH medium. The proportionality constants ǫ and µ are
known as permittivity and permeability of the medium respectively. Permittivity constant, ǫ0 , for the vacuum is approximately (1/36π) × 10−9 and
the permeability, µ0 , for the vacuum is equal to 4π × 10−7 in SI units. The
quantity, ǫr = ǫ/ǫ0 is called dielectric constant or the relative permittivity of
NPTEL Course: Wave Propagation in Continuous Media
3 Maxwell’s equations and material equations
4
the medium. The dielectric constant usually depends on temperature of the
medium. In case of alternating fields it also depends on the frequency of the
applied field. Materials with permittivity greater than that of the vacuum,
i.e. ǫ > ǫ0 , are called dielectrics whereas a material having permeability less
than that of the vacuum, µ < µ0 , is known as a diamagnetic. Paramagnetic substances are those for which, µ > µ0 . Paramagnetic substances(like
Magnesium, Manganese, Oxygen, Platinum) are attracted towards the high
magnetic field whereas the diamagnetic materials (like Antimony, Bismuth,
Carbon/Copper) are repelled by the magnetic fields.
Problem 1: Find the dimensions of ǫ and µ in SI units.
The third equation, (3.7), above is known as Ohm’s law, where the current
density is assumed to be proportional to the electric field.
Problem 2: Show the familiar form of the Ohm’s law (i.e. V=IR, where V
is the voltage across the sample, I is the current through the sample and R
is the resistance of the sample) by taking a sample as a straight wire of some
length with some cross-section.
If we take the divergence of the equation (3.4), we have
∇ · (∇ × H) = ∇ · j +
∂ρ
= 0,
∂t
(3.8)
where we have used the equation (3.1) and the fact that the divergence of
a curl vanishes. This is the continuity equation and it shows the charge
NPTEL Course: Wave Propagation in Continuous Media
3.3 The wave equation
5
conservation. With the materials equations now all the equations can be
written in terms E and B and can be solved with the constraint (3.3).
3.3 The wave equation
In the subsequent chapters we shall be mostly interested in the electromagnetic waves. We shall restrict ourselves to the source free regions, i.e. ρ = 0
and j = 0, unless otherwise specifically mentioned. With D = ǫE and
H = B/µ we have the source free Maxwell’s equations:
∇ · E = 0,
∇×E=−
(3.9)
∂B
,
∂t
∇ · B = 0,
∇ × B = µǫ
(3.10)
(3.11)
∂E
.
∂t
(3.12)
The interesting part of the electromagnetic field is that even for vanishing
sources it has non trivial solutions, which are known as electromagnetic waves.
Now if we take the curl of the equation (3.10) we have,
∇ × (∇ × E) = ∇(∇ · E) − ∇2 E = −
∂(∇ × B)
.
∂t
(3.13)
Now we use equations (3.9) and (3.12) to obtain the wave equation for the
electric field,
∂ 2E
1 ∂ 2E
2
µǫ 2 − ∇ E = 2 2 − ∇2 E = E = 0,
∂t
c ∂t
(3.14)
NPTEL Course: Wave Propagation in Continuous Media
3 Maxwell’s equations and material equations
6
√
√
where c = 1/ µǫ (for vacuum c = 1/ µ0 ǫ0 ≡ 3 × 108 m/s) is the speed of
the electromagnetic wave in the medium and the operator is known as d’
Alembertian.
Problem 3: Obtain the wave equation for the magnetic field, i.e. B = 0.
In the next chapter we shall find the solutions of these wave equations which
will be nothing but the electromagnetic waves.
3.4 Vector and scalar potentials
Potentials for the fields are defined in such a manner that their space gradients determine the electric and magnetic fields uniquely and they are also
consistent with the Maxwell’s equations. The form of the equation (3.3) suggests that one can always define a vector potential, A, such that curl of which
will produce the magnetic field B,
B = ∇ × A.
(3.15)
Note that divergence of B will be vanishing by the vector identity mentioned
above. Substituting the vector potential in the equation (3.2), we have with
a little rearrangement
∇ × (E +
∂A
) = 0.
∂t
NPTEL Course: Wave Propagation in Continuous Media
(3.16)
3.4 Vector and scalar potentials
7
Now one can use the vector identity ‘curl of gradient vanishing’ to define a
scalar potential, φ, such that
E+
∂A
= −∇φ,
∂t
(3.17)
so that the electric field is given by,
E=−
∂A
− ∇φ.
∂t
(3.18)
Problem 4: Eliminate E and B using (3.15) and (3.18) from Maxwell’s
equations and derive the equations for scalar and vector potentials.
Answers 4:
ρ
∂(∇ · A)
=− ,
and
∂t
ǫ
1 ∂ 2A
1
∂φ
= µj.
− ∇2 A + ∇ ∇ · A + 2
2
2
c ∂t
c ∂t
∇2 φ +
(3.19)
(3.20)
Problem 5: Show that if we change A by A + ∇Λ and φ by φ − ∂Λ
∂t (Λ being
a scalar function of space-time) the E and B remain unchanged hence the
scalar and vector potentials are not unique for a given electromagnetic field.
The above freedom in the choice of scalar and vector potentials is known
as the gauge degree of freedom. Usually one fixes this freedom either by
choosing ∇ · A = 0 (Coulomb, radiation or transverse gauge) or by setting
∇·A+
1 ∂φ
c2 ∂t
= 0 (Lorentz gauge). Using the latter condition the potential
equations become inhomogeneous wave equations i.e. equations with source
terms, φ = ρ/ǫ and A = µj.
NPTEL Course: Wave Propagation in Continuous Media
8
3 Maxwell’s equations and material equations
Problem 6: Show that Lorentz condition does not fix the gauge completely:
the potentials are not determined uniquely. Two different sets of potentials,
both satisfying Lorentz condition can produce the same electromagnetic field.
NPTEL Course: Wave Propagation in Continuous Media