Chapter Ten

10. Maxwell’s Equations

 Course Outline
10. Maxwell’s Equations (4 hrs)
10.1) Maxwell’s equations
10.2) Electromagnetic waves

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 10.1 Maxwell’s Equations
• Maxwell’s equations represent the laws of electricity and magnetism
that we have already discussed, but they have additional important
consequences.
• The four fundamental equations of electromagnetism, called
Maxwell’s equations and
• For simplicity, we present Maxwell’s equations as applied to free
space, that is, in the absence of any dielectric or magnetic material.
The four equations are

Gauss’s law (1 )

Gauss’s law in magnetism (2)

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 Faraday’s law (3)

Ampère–Maxwell law (4)

• Equation 1 is Gauss’s law: the total electric flux

through any closed surface equals the net charge
inside that surface divided by ε0.
Equation 2, which can be considered Gauss’s law in
magnetism, states that the net magnetic flux through
a closed surface is zero.

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• Equation 3 is Faraday’s law of induction, which
states that the emf, which is the line integral of the
electric field around any closed path, equals the
rate of change of magnetic flux through any
surface area bounded by that path.
• Equation 4, usually called the Ampère–Maxwell
law, is the line integral of the magnetic field
around any closed path is the sum of μ0 times the
net current through that path and ε0μ0 times the
rate of change of electric flux through any surface
bounded by that path.

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 10.2 Plane Electromagnetic Waves
• A combination of electric and magnetic field that travels through
space is an electromagnetic wave, or EM wave
• EMW Energy is a wave of energy with a frequency within the
electromagnetic spectrum, generated by the periodic fluctuation of an
electromagnetic field resulting from the acceleration or oscillation of an
electric charge
• An electromagnetic wave consists of oscillating electric and
magnetic fields.
• The properties of electromagnetic waves can be deduced from
Maxwell’s equations.
• One approach to deriving these properties is to solve the second-
order differential equation obtained from Maxwell’s third and fourth
equations.
• An electromagnetic wave traveling along an x axis has an electric
field and a magnetic field with magnitudes that depend on x and t:
• In this wave, the electric field E is in the y direction, and the
magnetic field B is in the z direction, as shown in the Figure below
.
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 • To generate the prediction of plane electromagnetic waves, we start
with Faraday’s law,

(5)

•To apply this equation to the wave in the above Figure , consider a
rectangle of width dx and height l , lying in the xy plane as shown in the
Figure below
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• The electric field in the y direction varies from E(x) to E( x + dx).
• We can express the electric field on the right side of the rectangle as

• Where E(x) is the field on the left side of the rectangle.

• Therefore, the line integral over this rectangle is approximately

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• Because the magnetic field is in the z direction, the
magnetic flux through the rectangle of area ℓdx is
approximately (ФB = Bℓdx. (This assumes that dx is very
small compared with the wavelength of the wave.) Taking
the time derivative of the magnetic flux, gives

• Substituting Equations 6 and 7 into Equation 3, gives

8
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• In the same manner, the line integral of B.ds is evaluated around a
rectangle lying in the xz plane and having width dx and length ℓ, as
in the Figure bellow.

• The magnetic field in the z direction varies from B(x) to B(x +dx)
• The line integral over this rectangle is found to be approximately

9
• The electric flux through the rectangle is ФE =Eℓdx, when
differentiated with respect to time, gives
10
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• Substituting Equations 9 and 10 into Equation 4 at I = 0, gives

11

• Taking the derivative of Equation 8 with respect to x and combining

the with equation 11, we get

• In the same manner, taking the derivative of Equation 8 with respect

to x and combining it with Equation 6, we obtain

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13 10
• Equations 11 and 12 both have the form of the general
wave equation with the wave speed v replaced by c,
where
(14)

• Where in
equation 10, we find that . Because this
speed is precisely the same as the speed of light in empty
space
• The simplest solution to Equations 12 and 13 is a
sinusoidal wave, we get
(15)
00000000000000000000 (16)
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• Where Emax and Bmax are the maximum values of the
fields. The angular wave number is k =2π/λ, where λ is the
wavelength. The angular frequency is ω=2πf, where f is
the wave frequency. The ratio ω/k equals the speed of an
electromagnetic wave, c :

• For electromagnetic waves, the wavelength and frequency

of these waves are related by
(17)
• Taking partial derivatives of Equations 15 (with respect
to x) and 16 (with respect to t), we find

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• Substituting these results into Equation 8, we find that at
any instant

• Using these results together with Equations 14 and 15, we

see that
(18)

• That is, at every instant the ratio of the magnitude of

the electric field to the magnitude of the magnetic field
in an electromagnetic wave equals the speed of light.

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• Let us summarize the properties of electromagnetic waves
as we have described them:
• The solutions of Maxwell’s third and fourth equations are
wave-like, with both E and B satisfying a wave equation.
• Electromagnetic waves travel through empty space at the
speed of light
• The components of the electric and magnetic fields of plane
electromagnetic waves are perpendicular to each other and
perpendicular to the direction of wave propagation. We can
summarize the latter property by saying that
electromagnetic waves are transverse waves.
• The magnitudes of E and B in empty space are related by
the expression E/B = c.
• Electromagnetic
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waves obey the principle of superposition.
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Example:
1.An electromagnetic wave in vacuum has an electric field
amplitude of 220 V/m. Calculate the amplitude of the
corresponding magnetic field.
Solution

2.In SI units, the electric field in an electromagnetic wave is

described by

Find
(a)the amplitude of the corresponding magnetic field oscillations,
(b) the wavelength λ , and
(c) the frequency f.
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 10. 3 Energy Carried by Electromagnetic Waves
• Electromagnetic waves carry energy, and as they
propagate through space they can transfer energy to
objects placed in their path.
• The rate of flow of energy in an electromagnetic wave is
described by a vector S, called the Poynting vector,
which is defined by the expression
(19) Poynting vector

• The magnitude of the Poynting vector represents the rate

at which energy flows through a unit surface area
perpendicular to the direction of wave propagation.
• Thus, the magnitude of the Poynting vector represents
power per unit area.
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• The direction of the vector is along the direction of
wave propagation as shown in the Figure bellow

• The SI units of the Poynting vector are

• The magnitude of S for a plane electromagnetic wave
where │ExB│= EB. In this case,

(20)
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• Because B = E/c, we can also express this as

• What is of greater interest for a sinusoidal plane

electromagnetic wave is the time average of S over one or
more cycles, which is called the wave intensity I.
• Hence, the average value of S (in other words, the
intensity of the wave) is
(21)
• the instantaneous energy density uE associated with an
electric field, is given by

• And the instantaneous energy density uB associated with a

magnetic field is given by
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• When we use the relationships B=E/c and the
expression for uB becomes

• Comparing this result with the expression for uE, we see

that

• That is, the instantaneous energy density associated

with the magnetic field of an electromagnetic wave
equals the instantaneous energy density associated
with the electric field.
• The total instantaneous energy density u is equal to the
sum of the energy densities associated with the electric
and magnetic fields:
Total instantaneous energy
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density of an EMW
• When this total instantaneous energy density is averaged
over one or more cycles of an electromagnetic wave, we
again obtain a factor of 1/2.
Average energy density of
an electromagnetic wave

• Comparing this result with Equation 21 for the average

value of S, we see that

• In other words, the intensity of an electromagnetic wave

equals the average energy density multiplied by the
speed of light.

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Example:-
1. At one location on the Earth, the rms value of the magnetic field
caused by solar radiation is 1.80 μT. From this value calculate
(a) the rms electric field due to solar radiation,
(b) the average energy density of the solar component of
electromagnetic radiation at this location, and
(c) the average magnitude of the Poynting vector for the Sun’s
radiation.
Solution

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 10.4 Momentum and Radiation Pressure
• Electromagnetic waves transport linear momentum as
well as energy.
• It follows that, as this momentum is absorbed by some
surface, pressure is exerted on the surface.
• The total momentum p transported to the surface has a
magnitude
Momentum transported to a
(complete absorption) perfectly absorbing surface
• The pressure exerted on the surface is defined as force per
unit area F/A.
• Let us combine this with Newton’s second law:

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• the momentum transported to the surface by radiation

• We recognize (dU/dt)/A as the rate at which energy is arriving

at the surface per unit area, which is the magnitude of the
Poynting vector.
• Thus, the radiation pressure P exerted on the perfectly
absorbing surface is
Radiation pressure exerted on a
perfectly absorbing surface
• If the surface is a perfect reflector (such as a mirror) and
incidence is normal, then the momentum transported to the
surface in a time interval ∆t is twice of p = U/c.
• That is, the momentum transferred to the surface by the
incoming light is p = U/c, and that transferred by the reflected
light also is p = U/c. Therefore,
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(complete reflection) 23
• The radiation pressure exerted on a perfectly reflecting
surface for normal incidence of the wave is
Radiation pressure exerted on a
perfectly reflecting surface

Example:-
1. A 100-mW laser beam is reflected back upon itself by a
mirror. Calculate the force on the mirror.
Solution

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