Maxwell’s Equations

Phy 108 course

Zaid Bin Mahbub (ZBM)
DMP, SEPS, NSU
Gauss’ Law for Magnetic Fields

One end of the magnet is a source of the field (the field lines diverge from it) and
the other end is a sink of the field (the field lines converge toward it).

By convention, we call the source the north pole of the magnet and the sink the
south pole, and we say that the magnet, with its two poles, is an example of a
magnetic dipole.

Suppose we break a bar magnet into pieces the way we can break a piece of
chalk. We should, it seems, be able to isolate a single magnetic pole, called a
magnetic monopole.

However, we cannot—not even if we break the magnet down to its individual

atoms and then to its electrons and nuclei. Each fragment has a north pole and a
south pole. Thus: The simplest magnetic structure that can exist is a magnetic
dipole. Magnetic monopoles do not exist (as far as we know).
Gauss’ “Law”

The net number of field lines leaving a closed

surface is proportional to the total charge
enclosed by that surface.

Two charges, equal in magnitude but opposite in sign, and the field lines that
represent their net electric field.

Four Gaussian surfaces are shown in cross section.

Surface A encloses the positive charge.

Surface B encloses the negative charge.
Surface D encloses no charge.
Surface C encloses both charges and thus no net charge.
Gauss’ Law for Magnetic Fields

Gauss’ law for magnetic fields is a formal way of saying that magnetic
monopoles do not exist. The law asserts that the net magnetic flux through any
closed Gaussian surface is zero:

Φ𝐵 = ර 𝐵. 𝑑 𝑆Ԧ = 0

Gauss law for magnetic field

Gauss’ law for magnetic fields holds for structures more complicated than a
magnetic dipole, and it holds even if the Gaussian surface does not enclose the
entire structure.
Gaussian surface II near the bar magnet of Fig. encloses no poles, and we can
easily conclude that the net magnetic flux through it is zero.
Gaussian surface I is more difficult. It may seem to enclose only the north pole
of the magnet because it encloses the label N and not the label S. However, a
south pole must be associated with the lower boundary of the surface because
magnetic field lines enter the surface there. (The enclosed section is like one
piece of the broken bar magnet in Fig.) Thus, Gaussian surface I encloses a
magnetic dipole, and the net flux through the surface is zero.
Ampère–Maxwell Law

Ampère’s law relates the line integral of the magnetic field around
some closed curve to the current that passes through any surface
bounded by that curve:

ර 𝐵. 𝑑 𝑙Ԧ = 𝜇0 𝐼𝑙

Maxwell recognized a flaw in Ampère’s law. Figure shows two

different surfaces S1 and S2, and bounded by the same curve C which
encircles a current carrying wire that is connected to a capacitor
plate.

The current through surface S1 is I but no current exists through Two surfaces S1 and S2 bounded by the
surface S2 because the charge stops on the capacitor plate. Thus, same curve C. The current I passes
ambiguity exists in the phrase “the current through any surface through surface S1 but not through
bounded by the curve.” surface S2
Ampère–Maxwell Law

Such a problem arises when the current is not continuous.

Maxwell showed that the law can be generalized to include all situations
if the current in the equation is replaced by the sum of the current and
another term Id called Maxwell’s displacement current, defined as,
𝑑Φ
𝐼𝑑 = 𝜖0 𝐸
𝑑𝑡

where Φ𝐸 is the flux of the electric field through the same surface
bounded by the curve l.

ර 𝐵. 𝑑 𝑙Ԧ = 𝜇0 (𝐼 + 𝐼𝑑 )
Ampère–Maxwell Law

𝑑Φ𝐸
Maxwell’s displacement current, 𝐼𝑑 = 𝜖0
𝑑𝑡

where Φ𝐸 is the flux of the electric field through the same surface
bounded by the curve l.
𝑑Φ𝐸
Ԧ
ර 𝐵. 𝑑 𝑙 = 𝜇0 (𝐼 + 𝐼𝑑 ) = 𝜇0 𝐼 + 𝜇0 𝜖0
𝑑𝑡

When there is a current but no change in electric flux (such as with a wire
carrying a constant current), the second term on the right side of Eq. is
zero, and so it reduces to the Ampere’s law. When there is a change in
electric flux but no current (such as inside or outside the gap of a charging
capacitor), the first term on the right side of Eq. is zero, and so it reduces
to the Maxwell’s law of induction.
 (a) Before and (d) after the plates
are charged, there is no magnetic field.
ර 𝐵. 𝑑𝑙Ԧ = 𝜇0 𝐼𝑠 (b) During the charging, magnetic field
is created by both the real current and
ර 𝐵. 𝑑𝑙Ԧ = 𝜇0 (𝐼 + 𝐼𝑑 ) the (fictional) displacement current. (c)
The same right hand rule works for both
currents to give the direction of the
magnetic field.
Maxwell’s law of induction
Induced Electric fields and Induced Magnetic Fields

Ԧ
A changing magnetic flux induces an electric field, and we ended up with Faraday’s law of induction (ℰ = ‫𝐸 ׯ‬. 𝑑 𝑙)
in the form,
𝑑Φ𝐵
Ԧ
ර 𝐸. 𝑑 𝑙 = −
𝑑𝑡
Here 𝐸 is the electric field induced along a closed loop by the changing magnetic flux Φ𝐵 encircled by that loop.

Can a changing electric flux induce a magnetic field?

𝒅𝜱𝑬
ර 𝑩. 𝒅𝒍Ԧ = 𝝁𝟎 𝝐𝟎
𝒅𝒕
(Maxwell’s law of induction).
Here 𝐵 is the magnetic field induced along a closed loop by the changing electric flux Φ𝐸 encircled by that loop.
Maxwell’s law of induction

According to Faraday’s law, a changing magnetic flux produces an electric field whose line integral around
a closed curve is proportional to the rate of change of magnetic flux through any surface bounded by the
curve.

Maxwell’s modification of Ampère’s law shows that a changing electric flux produces a magnetic field
whose line integral around a curve is proportional to the rate of change of the electric flux.

We thus have the interesting reciprocal result that a changing magnetic field produces an electric field
(Faraday’s law) and a changing electric field produces a magnetic field (generalized form of Ampère’s law).
Maxwell’s Equations: integral form

Gauss’ law for electricity: 𝜱𝑬 = ‫𝑬 𝒔 ׯ‬. 𝒅𝑨 = 𝒒/𝝐𝟎

Relates net electric flux to net enclosed electric charge

Gauss’ law for magnetism: 𝜱𝑩 = ‫𝑩 𝒔 ׯ‬. 𝒅𝑨 = 𝟎

Relates net magnetic flux to net enclosed magnetic charge

𝒅𝜱
Faraday's law: ‫𝑬 𝒄 ׯ‬. 𝒅𝑨 = − 𝒅𝒕𝑩
Relates induced electric field to changing magnetic flux

𝒅𝜱
Ampère-Maxwell law: ‫𝑩 𝒄 ׯ‬. 𝒅𝑨 = 𝝁𝟎 (𝑰 + 𝑰𝒅 ) = 𝝁𝟎 𝑰 + 𝝁𝟎 𝝐𝟎 𝒅𝒕𝑬
Relates induced magnetic field to changing electric flux and to current

* Written on the assumption that no dielectric or magnetic materials are present.

** In all four of Maxwell’s equations, the integration paths C and the integration surfaces A are at rest and the
integrations take place at an instant in time.
Maxwell’s Equations

Gauss’s law of electricity states that the flux of the electric field through any closed surface equals multiplied
by the net charge inside the surface.

This law implies that the electric field due to a point charge varies inversely as the square of the distance
from the charge.
This law describes how electric field lines diverge from a positive charge and converge on a negative charge.
Its experimental basis is Coulomb’s law.

Gauss’s law for magnetism states that the flux of the magnetic field through any closed surface is zero.

This equation describes the experimental observation that magnetic field lines do not diverge from any
point in space or converge to any point in space; that is, it implies that isolated magnetic poles do not exist.
Maxwell’s Equations

Faraday’s law states that the line integral of the electric field around any closed curve equals the
negative of the rate of change of the flux of the magnetic field through any surface bounded by curve
(is not a closed surface, so the magnetic flux through is not necessarily zero.)

Faraday’s law describes how electric field lines encircle any area through which the magnetic flux is
changing, and it relates the electric field vector to the rate of change of the magnetic field vector

Ampère’s law modified to include Maxwell’s displacement current states that the line integral of the
magnetic field around any closed curve equals multiplied by the sum of the current through any
surface bounded by the curve and the displacement current through the same surface.

This law describes how the magnetic field lines encircle an area through which a current or a
displacement current is passing.
 Maxwell’s Equations: Differential form

Gauss’ law (for electricity): Electrical charges are the source of the electric field

𝝆
𝛁. 𝑬 =
𝝐
Gauss’ law (for magnetism): There are no magnetic monopoles and the magnetic field lines can only circulate

𝛁. 𝑩 = 𝟎

Faraday’s law: The Curl of the electric field is caused by changing magnetic fields. A changing magnetic field
can produce electric fields with field lines that close on themselves

𝝏𝑩
𝛁×𝑬=−
𝝏𝒕

Ampère-Maxwell law: The Curl of the magnetic field is caused by current of charged particles (𝐽) or of the field
they produce (𝑑𝐸/𝑑𝑡). The strength of the field depends on the material
𝝏𝑬
Ԧ
𝛁 × 𝑩 = 𝝁𝑱 + 𝝁𝝐
𝝏𝒕
Two other vector fields are required when describing propagation of electromagnetism through matter
(rather than through vacuum): the electric displacement 𝑫 and the magnetic field intensity 𝑯 (also called
the “magnetizing force” or the “auxiliary field”).

We assume that any material is linear, isotropic, and homogeneous.

𝑩
𝑫 = 𝝐𝑬 and 𝑯 =
𝝁

where 𝜀 and 𝜇 are the electric permittivity and magnetic permeability of the material, respectively.

These are measures of the ability of the electric and magnetic fields to “permeate” the medium; if 𝜀 is
increased, then a larger electric field exists within the material, if 𝜇 is larger, then the magnetic field
intensity 𝐻 does not penetrate as far into the medium.
 Maxwell’s Equations: Differential form

𝝆 𝛁. 𝑫 = 𝝆
𝛁. 𝑬 =
𝝐 𝛁. 𝑩 = 𝟎
𝛁. 𝑩 = 𝟎
𝝏𝑯
𝝏𝑩 𝛁 × 𝑬 = −𝝁
𝛁×𝑬= − 𝝏𝒕
𝝏𝒕 𝝏𝑫
𝝏𝑬 Ԧ
𝛁×𝑯=𝑱+
𝛁 × 𝑩 = 𝝁𝑱Ԧ + 𝝁𝝐 𝝏𝒕
𝝏𝒕

Maxwell’s Equations: Integral form

‫𝑬 𝒔 ׯ‬. 𝒅𝑨 = 𝒒/𝝐𝟎 ර 𝒔 𝑫. 𝒅𝑨 = ම 𝝆 𝒅𝑽
‫𝑩 𝒔 ׯ‬. 𝒅𝑨 = 𝟎
𝒅𝜱𝑩 ර 𝒔 𝑩. 𝒅𝑨 = 𝟎
ර 𝒄 𝑬. 𝒅𝑨 = −
𝒅𝒕
𝒅𝑩
𝒅𝜱𝑬 ර 𝒄 𝑬. 𝒅𝑨 = − ඵ . 𝒅𝑨
ර 𝒄 𝑩. 𝒅𝑨 = 𝝁𝟎 (𝑰 + 𝑰𝒅 ) = 𝝁𝟎 𝑰 + 𝝁𝟎 𝝐𝟎 𝒅𝒕
𝒅𝒕
𝝏𝑫
ර 𝒄 𝑯. 𝒅𝑨 = ඵ 𝑱Ԧ. 𝒅𝑨 + ඵ . 𝒅𝑨
𝝏𝒕
Electromagnetic
wave equation
Electromagnetic
wave equation
𝑬 = 𝑬𝒎 𝐬𝐢𝐧(𝒌𝒙 − 𝝎𝒕)

𝑩 = 𝑩𝒎 𝐬𝐢𝐧(𝒌𝒙 − 𝝎𝒕)
As is true for the refractive index n, the permittivity and permeability in matter are larger than in
vacuum, 𝜀 > 𝜀0, and 𝜇 > 𝜇0.
In fact (though we won’t discuss it in detail), 𝜀 and 𝜇 determine the phase velocity v and the refractive
𝑐 1
index n via: 𝑛 = and 𝑣 =
𝑣 𝜇𝜖
1
For free space, speed of light, 𝑐 =
𝜇0 𝜖0
𝑬
All electromagnetic waves, including visible light, have the same speed c in vacuum. 𝒄 =
𝑩
Here, Ey and Bz are “the same,” traveling along x axis
The Traveling Electromagnetic Wave

An arrangement for generating a traveling electromagnetic wave in the

shortwave radio region of the spectrum: an LC oscillator produces a sinusoidal
current in the antenna, which generates the wave. P is a distant point at which
a detector can monitor the wave traveling past it.
The human body, like any object, radiates infrared radiation, and the amount emitted
depends on the temperature of the body. Although infrared radiation cannot be seen by the
human eye, it can be detected by sensors. An ear thermometer, like the pyroelectric thermometer
shown in Figure 24.11, measures the body’s temperature by determining the
amount of infrared radiation that emanates from the eardrum and surrounding tissue
A sinusoidal electromagnetic wave of frequency 40.0 MHz travels in free space in the x
direction as in Figure
(A) Determine the wavelength and period of the wave.
(B) At some point and at some instant, the electric field has its maximum value of 750 N/C and
is directed along the y axis. Calculate the magnitude and direction of the magnetic field at this
position and time.
The electric field component of an electromagnetic wave traveling in a vacuum is given by 𝑬𝒚 = 𝑬𝟎 𝐬𝐢𝐧 (𝒌𝒙 − 𝝎𝒕),
where E0 = 300 V/m and k=107/m. What are the frequency of the oscillations (in Hz) and the direction of
propagation?

The electric field component of an electromagnetic plane wave traveling in a vacuum is given by 𝑬 = 𝑬𝟎 𝐬𝐢𝐧 (𝒌𝒙 + 𝝎𝒕)ෝ
𝒚,
where E0 = 300 V/m and k=107/m. What is the magnetic field component of the electromagnetic wave? What is the
frequency of the electromagnetic wave (in Hz)?