Physics module on Electrostatics and
Magnetostatics
ICT programme
S. Anantha Ramakrishna,
Department of Physics, Indian Institute of Technology Kanpur
Contents
1
Introduction to electromagnetism
1.1
Some numbers associated with electromagnetism
1.1.1 Electromagnetism in natural phenomena
1.1.2 Electromagnetic devices
1.2
The electric charge and the Coulomb law
1.3
The superposition principle
page 1
2
3
4
5
7
2
Mathematical essentials of Vectors and Vector Calculus
2.1
Scalars and Vectors
2.1.1 Multiplication with vectors
2.1.2 Representation of vectors in terms of specified vectors
2.1.3 Symmetries of vectors under transformations
2.1.4 Visualizing scalar functions of multiple variables
2.1.5 Visualizing vector functions
2.2
A brief review of Vector Calculus
2.2.1 Fundamental theorem of Gradients
2.2.2 Gauss’s Divergence theorem
2.2.3 Stoke’s Theorem
2.2.4 Helmholtz theorem
2.3
Curvilinear geometries and coordinates
2.3.1 The Cartesian coordinate system
2.3.2 Cylindrical coordinate system
2.3.3 Spherical coordinates
2.3.4 An orthogonal curvilinear coordinate system
2.4
The Dirac δ− function
8
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9
10
11
12
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13
14
15
15
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17
18
19
20
3
Static charges, electric field and electric potential
3.1
Concept of the electric field
3.1.1 Superposition of electric fields and Charge distributions
3.2
Gauss’s law of Electrostatics
3.3
The curl of the electric field
3.3.1 The electrostatic Potential
24
24
25
27
27
27
iv
Contents
3.4
3.5
4
5
6
Energy associated with charge distributions
3.4.1 Problems regarding energy of a point charge
Conductors and capacitance
3.5.1 A cavity within a conductor
3.5.2 Surface charge on a conductor
3.5.3 Surface force on a conductor
3.5.4 Energy of a capacitor
Calculating the Electric field and potential
4.1
Poisson and Laplace equations: Boundary Value problems
4.2
Boundary conditions on the electrostatic field across charged
surfaces
4.3
Uniqueness theorems
4.4
The method of images
4.4.1 A conducting infinite plane
4.4.2 A conducting sphere
4.5
Sorting out some important problems
4.5.1 A conducting sphere placed in a homogeneous electric
field
4.5.2 Screening: fields within a atom
4.5.3 Why is a colloid suspension stable ?
4.5.4 Oscillations of a plasma
27
27
27
27
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28
28
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28
28
28
28
28
28
28
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Approximate description of the electric field at far-off points
5.1
Multipole expansions
5.2
Fields and potential of dipoles
5.3
Force, torque and energy associated with a dipole in electrostatic fields
29
29
29
Electrostatics of material media
6.1
Ideas of homogenization of the electric field in a material
medium
6.2
Bound charges, polarizability and macroscopic polarization
6.3
Electric field inside a material medium and The displacement
field
6.4
Dielectric susceptibility and dielectric permittivity
6.5
The electrostatic energy inside a dielectric medium
6.6
Sorting out some important problems
6.6.1 The electric fields of a uniformly polarized dielectric
sphere
6.6.2 A dielectric sphere placed in a homogeneous electric
field
6.6.3 Making a composite material with high dielectric
permittivity
30
29
30
30
30
30
30
30
30
30
30
Contents
6.6.4
6.6.5
7
8
9
Fields of a point charge placed near a flat, large
dielectric medium
Placing a dielectric inside a capacitor
v
30
30
Magnetic fields
7.1
Biot–Savart law for the magnetic field
7.1.1 Experimental verification of Biot–Savart law
7.2
Calculating fields for simple current configurations
7.3
Current distributions and magnetic fields for distributed
currents
7.4
Forces between current carrying conductors
7.5
Divergence and curl of the magnetic field and Amperés Law
7.6
The magnetic vector potential
7.7
Boundary conditions on magnetic fields across current sheets
31
31
31
31
31
31
31
31
31
Forces on a charge in electric and magnetic fields: The Lorentz
force
8.1
The Cyclotron frequency
8.2
Motion in parallel electric and magnetic fields
8.3
Motion in crossed electric and magnetic fields
8.3.1 Confining and focussing charges
32
32
32
32
32
Magnetostatics of material media
9.1
Multipole expansions for the magnetic field
9.2
Magnetic field and magnetic vector potential of a magnetic
dipole
~ field in a material medium
9.3
Magnetic fields and the H
9.4
Basic ideas of para-, dia-, ferro- magnetic media
9.5
Magnetic fields of magnetized objects
33
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33
Notes
Author index
Subject index
35
37
38
1
Introduction to electromagnetism
Electromagnetic forces all prevalent in our daily lives and at least 70% of our sensory perceptions arise optically, i.e., come from electromagnetic waves. Thus, it
would not be an exaggeration to say that electromagnetic phenomena dominate
our lives, particularly have even fashioned our lifestyles in modern times where
electronic gadgetry is so ubiquitous. Thus, understanding electromagnetism is one
of the first tasks for a Scientist and engineer. This module on electrostatics and
magnetostatics, is the first part of a course on Electromagnetism with the second
part being devoted to time-varying electromagnetic fields, optics and an elementary introduction to quantum mechanics mechanics. This module will lay down the
fundamental principles and also the mathematical background that is essential to
understand the second part of the course. We will use SI units throughout for all
quantities.
In nature, there are four fundamental forces:
1. gravitational forces
2. electromagnetic forces
3. Weak nuclear forces
4. Strong nuclear forces
The student must have already encountered the gravitational force that occurs
between masses in a previous course. The electromagnetic forces occur due to
static and moving electric charges and are responsible for holding together atoms,
molecules, crystals and much of the structure that we see and interact with on a
daily basis. The strong and weak nuclear forces are responsible for the stability of
nuclei that comprise most of matter. Figure ?? tries to capture relationship between
these forces, the historical origin of the studies of electromagnetism, its applications
and the possible future of our knowledge of the universe.
All of electromagnetism is governed by four equations that involve the sources of
the electromagnetic fields – the charges and the currents. We will put them down
2
Introduction to electromagnetism
just to have our perspectives in sight:
~ =
∇·E
ρ
,
ε0
(1.1)
~ = 0,
∇·B
(1.2)
~
~ = − ∂B ,
∇×E
(1.3)
∂t
~
~ = µ0 J~ + ε0 µ0 ∂ E .
(1.4)
∇×E
∂t
~ is the electric field, B
~ is the magnetic field, ρ is the charge density in space,
Here E
J~ is the current density in space and the above equations describe the variation of
~
the electric and magnetic fields in space and time (t) for given sources (ρ and J).
The ∇ operator is described in Chapter ??, ε0 and µ0 are some constants that will
be described later. All of electrostatic and magnetostatic phenomena in this module
can be described by simply setting to zero the time derivatives in these equations as
there is no time dependence. These equations were originally written down by J.C
Maxwell, and hence they are called Maxwell’s equations or the Maxwell equations.
It is interesting to note that electromagnetism is the first fundamental force that
mankind discovered properly - the four Maxwell’s equations were automatically
Lorentz invariant and relativistically correct.
1.1 Some numbers associated with electromagnetism
In this section, I will present some numbers for electromagnetic quantities that are
associated with various pheomena of electromagnetism and electromagnetic devices.
Electromagnetic quantities that one is concerned with are typically the electric and
magnetic fields, chrages and currents that drive these fields, the frequencies at which
the electromagnetic fields vary at, energies associated with various effects and so
on. This discussion some particular examples will hopefully raise the curiosity to
understand the phenomena better and might enable the student to appreciate the
magnitudes of the numbers that (s)he might calculate in this course.
There is another picture of electromagnetism that is placed in Figure ??. This
picture shows a frequency scale for the electromagnetic spectrum indicating where
various electromagnetic phenomena happen. On the left, we have time-independent
or static phenomena that correspond to zero frequency. All of the discussion in this
module will concern that. Higher up we have radio-frequency waves that are so essential to communications from about hundreds of kilohertz to tens of megahertz.
(hertz is the unit of frequency (Hz being the symbol). Higher frequencies than these
are the microwaves, from hundreds of MHz to hundreds of gigahertz (GHz) which
have refashioned the way we live, from mobile phones, microwave ovens to radars
for airplanes and ships. At even higher frequencies, we have the terahertz (THz)
waves that form a part of the electromagnetic spectrum that have not been utilized
1.1 Some numbers associated with electromagnetism
3
by mankind very well so far, but has future promise for many applications in the
future using manmade materials (metamaterials) for purposes of detection, sensing and security scanning applications. Above the Terahertz, we have the infra-red
part of the spectrum that is so essential for spectroscopy of molecules by which
we have learnt so much about molecular structure. The near-infra-red part of the
spectrum (200 THz or 1500 nm wavelengths) have been very fruitfully utilized for
long distance communications through fibre optics. The uhman eye can sense only
a very small part of the spectrum called visible frequencies extending from 400 nm
to about 700 nm wavelengths (700 - 400 THz). The blackbody radiation of the sun
peaks in the yellow-green part of the spectrum and nature has beautifully utilized
it with the human eye being most sensitive at about 555nm wavelength. At higher
frequencies, we have the ultra-violet spectrum ( 1015 Hz), which most of the atomic
transitions emit, and X-rays at even higher frequencies ( THz) Both Ultra-violet
and X-rays have ionizing properties and exposure to them beyond certain safe limits
can result in cancers in our bodies. However, X-rays have proved essential for medical imaging of our bones and interior organs as well as for actually understanding
(imaging in a sense) the crystal structures of most of solid matter. Higher beyond
these frequencies, we have the Gamma (γ) rays that are emitting by nuclear reactions and the cosmic rays whose origins are not completely understood even today.
It must be emphasized that most of our knowledge about matter and this universe
has arisen from the detection of electromagnetic waves at various frequencies across
this spectrum.
1.1.1 Electromagnetism in natural phenomena
Let us consider some natural phenomena involving electromagnetism and talk about
the magnitudes of various electromagnetic quantities that are involved.
Earth’s atmosphere
It might surprise the student to to know that there is an almost uniform vertical
electric field at the earth’s surface of about 100 V/m. While the presence of buildings
etc. that are somewhat conducting will deform this electric field, this field is readily
detectible in open areas or over the surface of a large water body such as a lake or
the sea. This electric field extends to several kilometres although its magnitude does
reduce with increasing altitude because the higher regions of the atmosphere are
more conducting and tenuous. The total potential difference from the earth’s surface
to the top of the atmosphere is about 400,000 Volts, the earth being negatively
charged with respect to the atmosphere! Due to ionization caused by solar radiation
in the upper atmospheric stretches, currents flow in the atmosphere driven by this
potential difference. This integrated current over the earth’s surface is about 1800
amperes implying that the net energy associated with this process is about 700
megawatts, which is the ouput of a large power plant.
4
Introduction to electromagnetism
So let us ask, what is the reason for this electric field, for the earth’s negative
charge? The reason it turns out is that the lightning strikes during a thunderstorm are mostly negatively charged, thereby inducing a net negative charge on
the earth’s surface. The reason why lightning is, most of the times, negative is due
to the peculiar charge distribution of charges on a cloud that is mostly negative
in the lower stretches while being positive in the upper region. The real reason
for this peculiar charge distribution is however not known yet. The student is referred to Chapter -6 of Ref. ? for a more detailed description of atmospheric effects.
Lightning
Lightning is easily one of the most impressive natural phenomena. A large amount
of charge built up on clouds discharges to the earth in one sudden burst. The time
of the lightning flash is about 0.2 seconds in which a typical total charge of about
2.5 coulomb (it can range from 1 to 6 C) is discharged over a distance of about 2
km. This is a good place to point out how large the unit of charge is in SI units.
The peak currents are about 20 kA with an energy dissipation of about 100 kJ /m
along the path. An important parameter here is the voltage at which air breaks
down and becomes effectively a conductor. Initially a few ions present in the air
accelerate down the voltage column, and cause more ionization in the air due to
collisions with other atoms (avalanche process).This large density of ions effectively
creates a conducting column for lightning. Air at room temperature and pressure
breaks down when an electric field of about 30kV / cm is applied across the air column. This breakdown at a specific voltage has been utilized in making fast switches
(spark gaps) in automobile technology for a long time.
Earth’s magnetic fields
Neurons and electricity
Inside our bodies an enormous amount of information and signals are transmitted
via specialized cells called neurons. The signals are transmitted in between neurons
across spaces called neuronal synapses. The neuron is held at a resting potential
of about -70mV which can be switched off suddenly thereby transmitting a signal.
This rest potential ‘is accomplished by segregating ions such as Na+ , K+ and Ca2+
from negative chloride ions by semi-permeable membranes.
1.1.2 Electromagnetic devices
Now we will proceed to some modern devices that crucially depend on electromagnetism and the typical values of electromagnetic quantities that arise in them. One
should remember that devices and mechanisms being utilized for devices change
very quickly and the numbers can be quite different in the matter of few years.
Hence it would be important to keep in mind the year 2010 in which the present
1.2 The electric charge and the Coulomb law
5
section is being written.
Magnetic resonance imaging
Mobile Phones
X-ray machines
DC and induction motors
Magnetic memories
Semiconductor chips and computers
1.2 The electric charge and the Coulomb law
From various experiments, (that historically involve rubbing various insulators such
as glass rods and cat’s fur, or frictional electricity), it has been deduced that there
are two kinds of charges – positive and negative. The result of such experiments
can be summed up as that like charges repel each other while unlike charges attract
each other.
Usually it is thought that a neutral object is composed of equal amounts of positive and negative charges. At a more fundamental level, we know that fundamental
particles either have a positive (examples are the proton and the positron) or a
negative charge (examples are the electron, the antiproton, the muon etc.) or are
neutral (an example is the neutron). Modern Physics tells us that for every elementary particle there is an anti-particle that will have an equal mass but opposite
charge. For example, we have the positron that is the antiparticle for the electron,
the anti-proton for the proton and so on. We have even managed to make antihydrogen in 2003 by getting a positron into a bound state with an anti-proton! A
particle and its anti particle can annihilate and their energy (including rest mass) is
converted into a gamma ray with the appropriate energy. For example, an electron
annihilates with a positron:
e− + e+ = γ
(1.5)
where the gamma ray has a minimum energy of 1.02 MeV that corresponds to
2me c2 the rest mass of the electron and the positron. Note that the above equation
should conserve energy and momentum too.
In addition, in any such reaction the net charge should be conserved too. We
have experimentally found that the algebraic sum of the charges in an isolated
system is always constant. A stronger statement would be that the total charge is
6
Introduction to electromagnetism
conserved in any inertial system. Note that while notions such as length and time
intervals change under a Lorentz transformation, the total charge is conserved. A
much better justification of this claim is needed and will be done when we discuss
transformation of electromagnetic fields later.
Another important aspect about the charge is that it is quantized. It always
occurs in any experiment in multiples of the electronic charge. The electronic charge
in SI units is e = −1.609 × 10−19 coulomb. Millikan established with his famous oildrop experiment that even in macroscopic situations, the charge is always a multiple
of e. The charge of a proton and an electron are identical in magnitude upto 1 part
in 1020 (atoms are neutral). The electric force is such a strong force that any larger
discrepancy would be very easily discernible at terrestrial scales. It is now known
that particles (hadrons) such as the proton and the neutron have internal structure
and are made of particles called quarks. These quarks have fractional charge: the
up-quark has charge 2e/3 while the down-quark has −e/3. But quarks are never
observed as free particles. They are always present in bound states represented
other particles: for example, the proton is a bound state of two up-quarks and one
down quark resulting in a net charge of +e, while the neutron is a bound state of
one up-quark and two down-quarks resulting in a net zero charge for the neutron.
In electrostatics, the fundamental question is, given a set of charges located at
certain positions, what would be the force on a given charge at a given position?
At the heart to the solution of this question is an empirical law that governs the
forces (of repulsion or attraction) between two charges – the Coulomb’s law. This
law states that the force exerted by one point charge (of zero spatial extant) of
magnitude q1 on another charge of magnitude q2 is given by
q1 q2
F~ = k 2 r̂21
r21
(1.6)
where r21 is the distance from the charge q1 to the charge q2 . The force is directed
along the vector r̂21 from charge -1 to charge -2 and is inversely proportional to the
square of the distance between them. The force is attractive or repulsive depending
on the relative signs of the two charges.
In SI Units, the numerical value of the proportionality constant k depends on the
units used. In the SI Units,
k=
1
= 8.9 × 109 N · m2 /C2 .
4πε0
(1.7)
The Coulomb that is the unit for the charge in SI units is a very large unit. However,
it is very convenient to define the units of current as ampere = coulomb per second,
which seems to be a very convenient measure for the current in most everyday
situations and is hence retained in the SI system of units.
The validity of Coulomb’s law has been subjected to intense scrutiny The inverse
square behaviour with the charge separation distances appears almost exact. One
1.3 The superposition principle
7
may write
F ∝ r−(2+)
where is the deviation from the inverse square behaviour. Experimentally, one
may fix limits on the maximum magnitude of , depending on the sensitivity and
accuracy of the experiment. This has been a cause for concern since the times of
Cavendish who found that < 10−2 and Maxwell ( < 10−6 ) down to modern times
where modern experiments based on variants of the original Cavendish experiment
have obtained < 10−14 and from geomagnetic measurements ( < 10−16 ). Today, we believe that the Coulomb force obeys an inverse square behaviour exactly.
Any deviation from an exact inverse square behaviour would have serious repurcussions: for example, the photon would then have a non-zero photon rest mass.
Another amazing aspect of the Coulomb law is the range of lengthscales where it
has been tested and found valid. We have confirmed the Coulomb force law down
to lengthscales of 10−15 m (at lower lengthscales we have electro-weak corrections)
while measurements on the magnetic field of jupiter have confirmed this law to the
large lengthscales of 108 m. Thus, it can be said that we have enormous confidence
in this law.
1.3 The superposition principle
The coulomb’s law of electrostatics describes the force exerted by one charge upon
another. However, what would be the force on a given charge at a given location
due to several charges ?
It has been found that the electrostatic force exerted on one charge by another
is independent of the presence of a third charge. Also, it has been found that the
electrostatic forces on any one charge due to several others add up vectorially. Thus,
the net force on charge -1 due to other charges (2, 3, · · · ) is given by
F~1 = F~12 + F~13 + F~14 + · · ·
(1.8)
The superposition principle is a profound principle of Physics without which it
might have been almost impossible to calculate anything in electromagnetism. Imagine the complicated situation that would result if the presence of a third charge
changed the interaction of two charges. such situations can arise with some nonlinear systems. But linear superposition works at a fundamental level with electric
and magnetic fields.
2
Mathematical essentials of Vectors and Vector
Calculus
Here, we will first introduce the reader to the basic Mathematical knowledge and
obtain some essential Mathematical skills necessary to calculate and deal with Electric and magnetic fields of more complicated charge and current distributions. Since
these fields are essentially are vectorial in nature and the equations that we will find
to describe them involve derivatives of these quantities, we will quickly go through
the fundamental theories and results of vectors and vector Calculus. The student
is advised to work through this chapter in order to rigorously follow the rest of the
material.
2.1 Scalars and Vectors
Most students are aware that there are quantities such as the mass of an object,
the distance covered by a car in a year (look at the odometer), temperature etc.
that can be associated with a number and there are physical quantities such as the
displacement of an object, or the velocity of a moving car that not only have a
number , but also a sense of direction associated with them. Further, they do not
add up like ordinary numbers. For example, if you went south for 4 kms and east
for 3 kms, the net displacement from the initial would be 5 km. Further, the final
location is not uniquely specified unless the direction from the initial point is also
stated. However, we would need to make more rigorous these intuitive ideas.
~ is an abstract object
Mathematically, a vector (denoted by an overhead arrow A)
that belongs to a set. The essential conditions are that
1. You can associate by a given rule (add) any two of these vectors to produce
~ =A
~ + B,
~ where ~a, B,
~ C
~ belong
another vector that belongs to the same set: C
to the set.
2. Further, this association (addition) does not depend on the order that you con~+B
~ =B
~ + A.
~
sider them (addition is commutative): A
3. There exists an identity (zero) vector (~0 in the set which when associated with
~ + ~0 = A.
~
any vector in the set produces the same given vector: A
2.1 Scalars and Vectors
9
4. For every vector in the set, there exists another vector in the set, such that they
~ + (−A)
~ = ~0.
add up to the zero vector: A
An example of the above mathematical objects are the real vectors in three
dimensional space that can be represented by arrows, with the length of the arrows
denoting the magnitude of the vector and the orientation denoting the direction.
The Zero vector has zero magnitude or length. When we add two such vectors,
we imply that the tail of one vector must be placed at the head of the other (as
shown in Fig. ??) with the resulting vector being represented by the arrow that
joins the tail of the latter to the head of the former vector. Familiar examples of
such quantities are the displacement, velocity, forces etc.
2.1.1 Multiplication with vectors
We can envision the following forms of multiplication involving vectors:
1. Multiplication of a vector by a scalar.
Multiplication of a vector by a scalar gives a vector in the same direction and
a different magnitude. For real vectors in the three dimensional space, we will
constrain the scalar to be real numbers. Then the resulting vector is thought
to remain along the same direction as the original vector, but with a different
length.
a . ~v = a~v ,
where a is a scalar and ~v and a~v are vectors. For real vectors, the resulting
vector’s length is equal to the length of the original vector multiplied by the
modulus of the scalar number.
2. Multiplication of two vectors to yield a scalar.
We can associate a unique scalar number by a given rule with every two pair of
vectors. For real vectors, we can use the rule
~a · ~b = ab cos θ
where a, b are the lengths of the two vectors ~a and ~b respectively and θ is the
angle between the vectors. ~a · ~b is called the scalar product. The dot product is
commutative, and distributive, i.e.,
~a · ~b = ~b · ~a~a · (~b + ~c) = ~a · ~b + ~a · ~c
3. Multiplication of two vectors to yield a vector This is defined for two real vectors
as
~a × ~b = ab sin θ.
(2.1)
This is called a cross product and is denoted by the symbol ×. The resultant
is a vector. The magnitude of the cross product corresponds to the area of
10
Mathematical essentials of Vectors and Vector Calculus
parallelogram whose sides are defined by the two vectors and the direction is
defined to be perpendicular to the parallelogram. It is obvious that the cross
product is zero if either of the two vectors are zero or if they are parallel. It is
common to utilize the cross product to define areas using the above idea.
The cross product is anti-commutative,
~a × ~b = −~b × ~a,
and distributive
~a × (~b + ~c) = ~a × ~b + ~a × ~c.
Note that the associative property does not hold true. i.e.,
~a × (~b × ~c 6= (~a × ~b)~c
4. Scalar triple product.
Another frequently appearing quantity is the triple product defined by
[~a, ~b, ~c] = ~a · (~b × ~c).
(2.2)
The value of this quantity corresponds to the volume bounded by the parallelogram whose sides are defined by the three vectors. It is easy to show that the
following triple products are equal
[~a, ~b, ~c] = [~c, ~a, ~b] = [~b, ~c, ~a],
i.e., any cyclic permutation of the order of the vectors yields the same triple
product.
5. Vector triple product.
It can be shown that
~a × (~b × ~c = ~b(~a · ~c) − ~c(~a · ~b)
Use the cartesian representation of vectors to prove this (see below). This very
useful identity can easily be remembered by noting it as the BAC-CAB rule.
2.1.2 Representation of vectors in terms of specified vectors
A vector can be represented in terms of certain specified vectors. For example, we
can prefer to think of a given displacement, for example, from Kanpur to Jhansi
as x kilometres west and y kilometres south. Then we can write the displacement
vector as ~v = xx̂ + y ŷ, where the unit vectors x̂ and ŷ represent the directions
along east and the north. The choice of the unit vectors is usually along orthogonal
directions for the sake of convenience, but it is not necessary that the unit vectors
be orthogonal. In three dimensional space, we will denote the unit vectors in the
2.1 Scalars and Vectors
11
three orthogonal directions as x̂, ŷ and ẑ. Any given vector can be resolved into
components along these unit vectors
~ = Ax x̂ + Ay ŷ + Az ẑ.
A
(2.3)
The components of a given vector are obtained by the scalar products with the
respective unit vectors:
~
Ax = x̂ · A,
~
Ay = ŷ · A,
~
Az = ẑ · A.
(2.4)
2.1.3 Symmetries of vectors under transformations
We start by asking how do vectors transform under various transformations? What
are the physical properties that essentially distinguish between vectors and scalars
? Scalars have the same value regardless of how we view them – thus they are
invariant under physical transformations. However consider, for example, if any
arbitrary ordered triplet such as (n, m, l) where n represents the number of apples
in a barrel, m the number of oranges and l the number of bananas, be considered as
a vector? Clearly such a quantity remains a barrel with the same number of apples,
oranges and bananas. Let us understand how real vectors transform in contrast
under the following physical transformations:
1. Shift of origin: Clearly a vector is unaffected in direction and magnitude by a
shift of the origin. Hence the transformed vector ~v 0 = ~v . The vector can be
shifted around while keeping the direction and magnitude unaffected.
2. A rotation of the coordinate system: Let us consider vectors in two dimensions
for clarity and a rotation of the axes (X − −Y axes about the Z axis) by an
angle φ as shown in Fig. ??. Clearly, the Cartesian components in the new set
of axes are related to the old components as
x0 = x cos φ + y sin φ
(2.5)
0
(2.6)
y = −x sin φ + y cos φ
Thus, writing the vectors as column matrices,
0 x
x
0
~v =
and ~v =
y0
y
, the transformed vector is given by
~v 0 = R~v
where
R=
cos φ
− sin φ
sin φ
cos φ
(2.7)
12
Mathematical essentials of Vectors and Vector Calculus
is the rotation matrix. The vecv can be obtained from ~v 0 via the inverse matrix
R−1 . Clearly the manner in which the components of a vector combine to produce
the components in the rotated coordinate system is very different from the way
apples and oranges behave in our above example.
3. Inversion of the coordinates Imagine if we reflected each point by a mirror placed
on the principal planes (X − Y plane, Y − Z plane and the Z − X plane)
containing the origin and assigned to each point in the new coordinate system,
the coordinates of the reflected point:
x → −x,
y → −y,
z → −z.
It is obvious that each vector in the transformed system is the negative of the
original vector ~v 0 = −~v .
Note that objects such as the cross product ~a = ~b×~c remain invariant under an
inversion – hence they are called pseudo-vectors. Similarly, scalar triple product
changes sign under an inversion – hence they are called pseudo-scalars.
4. Time reversal: t → −t.
This is another symmetry operation that is important in Physics, particularly
when analyzing the dynamical nature of systems. Objects such as the velocity
(the time derivative of the displacement) pickup a negative sign under the time
reversal transformation.
2.1.4 Visualizing scalar functions of multiple variables
It is important to be able to visualize functions of multiple variable that we will continuously encounter in our study, for example, the electric potential that depends
on the three Cartesian coordinates of the point. In many case, just by being able to
visualize the function we may be able to obtain an deep insight into the behaviour
of the physical system. If we have a function of two other variables, we can project
the function along one axis and visualize the behaviour of the function as shown
in Fig ??. An alternative mode if the number of independent variables were three
would be give a shading or color coding as shown in Fig ??. The student is encouraged to picture these functions and to learn the use of computer softwares such as
MATLAB or MATHEMATICA that would enable him/her with the capability for
visualization of quite complicated functions.
2.1.5 Visualizing vector functions
It is clear that for representing vector functions, one would need to represent the
direction as well as the magnitude at every point. One typically uses arrows at
representative points to do this: the length or thickness of the arrows at each point
could serve to indicate the magnitude with the arrow directed along the vector at the
2.2 A brief review of Vector Calculus
13
given point (an example being shown in Fig. ??). Another possible representation
of this could be to draw streamlines with the tangent to the streamline being along
the vector field at each point. The density of the streamlines would have to be
proportional to the magnitude of the field (an example is shown in Fig ??) with
colour being used as another possible label for the magnitude of the field.
2.2 A brief review of Vector Calculus
In this section, we will briefly describe some essential theorems of vector calculus that we will use in our description of electrodynamics. The student is urged
to develop an acquaintance with these results as we will repeatedly require them
throughout our discussions. The proofs of the theorems will, however, not be presented here, and the student is referred to Ref. ?? for the proofs.
Consider a continuous function f (x, y, z). The infinitesimal change in the function
due a a infinitesimal change in the position: from (x, y, z) to (x + dx, y + dy, z + dz)
is given by
∂f
∂f
∂f
dx +
dy +
dz
(2.8)
df =
∂x
∂y
∂z
where (∂f )/(∂x) etc. are the partial derivatives of the function, i.e., describe the
derivative of the function due to a change of that variable only while keeping other
variables fixed. The above relation is the fundamental relation of differential calculus. Noting that the infinitesimal shift in the position can be described by the
infinitesimal vector d`, we write
∂f
∂f
∂f
x̂ +
ŷ +
ẑ · (x̂dx + ŷdy + ẑdz)
df =
∂x
∂y
∂z
~
= ∇f · d`
(2.9)
where the vector operator ∇ called the gradient and is defined to operate on a scalar
function such that
∂f
∂f
∂f
∇f =
x̂ +
ŷ +
ẑ.
(2.10)
∂x
∂y
∂z
Since the change of the function would be maximum when the displacement is
parallel to the gradient, we realise that the vector ∇f points along the direction of
greatest increase in f and |∇f | gives the magnitude of the slope in this direction.
Hence the name gradient for this object.
We can define a vector operator ∇
∇ = x̂
∂
∂
∂
+ ŷ
+ ẑ ,
∂x
∂y
∂z
(2.11)
with the understanding that to be meaningful, the operator has to operate on a
function. Once this vector operator has been defined, we can define other operations
14
Mathematical essentials of Vectors and Vector Calculus
with it, such as operation on a vector field through a scalar product or through the
vector product. Operation through the scalar product yields the divergence:
~ y, z)
∇ · A(x,
∂f
∂
∂
+ ŷ
+ ẑ
· (Ax x̂ + Ay ŷ + Az ẑ),
= x̂
∂x
∂y
∂z
∂Ax
∂Ay
∂Az
=
+
+
.
∂x
∂y
∂z
(2.12)
~ through
In a similar manner, operation of the gradient operator on a vector field A
a vector product yields
x̂
ŷ
ẑ ~ = ∂/∂x ∂/∂y ∂/∂z (2.13)
∇×A
A
Ay
Az
x
∂Az
∂Ay
∂Ax
∂Az
∂Ay
∂Ax
= x̂
−
+ x̂
−
+ x̂
−
.
∂y
∂z
∂z
∂x
∂x
∂y
The divergence of a vector field is related to the sourrces or sinks of a vector field
while the curl is related to the rotational aspects of the vector field. These aspects
will become clear below when we discuss the Divergence and Stokes theorems.
EXAMPLE:
1. Calculate ∇( |~r−1r~0 | ) where r~0 is a fixed vector
∇(
1
|~r − r~0 |
)=
=
x̂
∂
∂
∂
+ ŷ
+ ẑ
∂x
∂y
∂z
[(x −
=−
x0 )2
1
0
2
[(x − x ) + (y − y 0 )2 + (z − z 0 )2 )]1/2
−(x − x0 )x̂
+ (y − term) + (z − term)
+ (y − y 0 )2 + (z − z 0 )2 )]3/2
(~r − r~0 )
|~r − r~0 |3
(2.14)
2. Obtain the divergence
3. Obtain the curl
2.2.1 Fundamental theorem of Gradients
The Fundamental theorem of Gradients states that the line integral of the gradient
of a scalar function f (~r depends only on the values of the end points (r~1 and r~2 ):
Z ~r2
∇f (~r) · d~r = f (r~2 ) − f (r~1 ).
(2.15)
~
r1
2.2 A brief review of Vector Calculus
15
This is consistent with the interpretation of the gradient that the infinitesimal
change in the function
df = ∇f · d~r. This also implies that the line integral over a
H
closed loop C is C ∇f (~r) · d~r = 0.
2.2.2 Gauss’s Divergence theorem
~ r) and states that
This important result concerns the divergence of a vector field A(~
net flux of a vector field out of a closed surface S enclosing a volume V (See Fig. ??)
is related to the volume integral of the divergence of the field:
Z
I
~ 3r =
~ · d~s.
(∇ · A)d
A
(2.16)
V
S
~ is expected to satisfy certain conditions of continuity that are usually
Note that A
resolved by most fields that describe physical quantities. If one thinks of the vector
field as describing the velocity flow of an incompressible fluid, then the divergence
theorem essentially states that the net efflux of the fluid through a closed surface
must come through sources of production of the fluid in the volume within the
surface. In fact, the very idea of the divergence as the source for the vector field
becomes obvious by taking the limit of an infinitesimal volume:
H
~ · d~s
A
~ = lim
.
(2.17)
∇·A
∆V →0
∆V
There is an interesting aspect to the Divergence theorem. The value of the volume
~ that should depend on the behaviour of the vector field in the
integral of ∇ · A
interior of the volume turns out to be entirely determined by only the values of
the field on the closed surface bounding the volume. This surprising fact arises
primarily due to the continuity of the field within the given volume.
2.2.3 Stoke’s Theorem
Another important and often used result is the Stokes theorem that concerns the
curl of a vector field. It relates the surface integral of the curl over an open surface
(S) to the line integral of the field over the closed loop, C, bounding the open
surface (See Fig. ??):
Z
I
~ · d~r.
~ r) · d~s =
A
(2.18)
∇ × A(~
S
C
Here, once again, the vector field is assumed to satisfy the minimal amounts of
continuity that are usually satisfied by fields representing most physical quantities.
The Stokes theorem also interestingly relates the value of a surface integral over
the curl of the vector field to the values of the vector field on the closed curve
enclosing the surface – thus, it is merely the values of the function on the loop C
that determine the integral. There is an important fact that should be pointed out:
16
Mathematical essentials of Vectors and Vector Calculus
the loop C contains an infinite number of possible surfaces. Think about a balloon
– the circular curve describing the mouth of the ballon can be taken to be C. Now a
stretched sheet across this mouth is a possible surface. Now if this balloon is blown
out to different shapes and sizes, each one of the surfaces is a possible surface that
is bound by the closed curve C. The really amazing thing about the Stokes theorem
is that it holds true for each and everyone of them – the flux of the curl through
any of them for a common bounding curve is identical.
2.2.4 Helmholtz theorem
~ r) in a given region of space V . The Helmholtz theorem
Consider a vector field A(~
states that if we can uniquely specify the divergence and curl of the vector field
everywhere within V , and additionally also specify the normal component of A on
~ is uniquely specified.
the bounding surface S of the volume V , then the field A
Following this theorem, we can separate the given vector field into two parts:
~ r) = A
~D + A
~R
A(~
(2.19)
~ D is an irrotational field with zero curl and non-zero divergence only, and
where A
~
~ = ∇·A
~ D and
AR is a divergenceless field with a non-zero curl only such that ∇ · A
~
~
∇ × A = ∇ × AR . The Helmholtz theorem is of a great help in those situations
where we know the divergence and the curl of a vector field. Then we can be sure
that there is a unique vector field that has the divergence and the curl, subject to
the specification of the boundary terms.
2.3 Curvilinear geometries and coordinates
In our discussions of electromagnetism in this course, very often we will deal with
geometries containing cylinders or spheres. The Cartesian coordinate geometry is
not the most well suited system to handle spherical and cylindrical geometries. Particularly, if there are symmetries associated with the problem such as an invariance
with angle or distance from a given point, considerable simplifications can occur
in the calculations if other coordinate systems are used. Usually it is simpler to
consider coordinate systems with orthogonal axes. Here we will formally introduce
and detail the three orthogonal coordinate systems that we will frequently use.
2.3.1 The Cartesian coordinate system
This is the familiar coordinate system to the student. Consider space in three
dimensions: let us choose one point and call it the Origin. Now choose three mutually
perpendicular axes in three dimensions that we will call as the X, Y, and the Z axes
that intersect at the Origin (see Fig. ??. We label every point in space by three
2.3 Curvilinear geometries and coordinates
17
numbers, (x, y, z), that correspond to the distances from the origin that one would
have to travel parallel to the three axes. Three unit vectors (x̂, ŷ, ẑ) are defined
in the directions along the three principal axes. Note that these unit vectors are
constant vectors and are the same when transposed to any given point – this follows
from the property that the principal surfaces are planes in this coordinate system
(see Fig. ??. By definition, we have x̂ · ŷ = ŷ · ẑ = ẑ · x̂ = 0.
We list the following quantities for the sake of completeness and comparison with
other coordinate systems:
1. The infinitesimal line element : d~r = x̂dx + ŷdy + ẑdz.
2. The infinitesimal volume element: d3 r = dx dy dz.
3. The infinitesimal surface elements: d~sx = dy dz x̂, d~sy = dz dxŷ, d~sz = dx dyẑ.
2.3.2 Cylindrical coordinate system
This becomes useful when the problem at hand has a preferred axis and when the
fields primarily depend only on the absolute distance of the point from the the
preferred axis. In this system, we label each point in space again by three numbers:
but only two of them correspond to distances while the third corresponds to an
angle. First we take the preferred axis (direction) and call it the Z axis. Choose
the origin on this axis, and arbitrarily choose another direction in the plane (–
that would be the X-Y plane) perpendicular to the Z axis. Now any point in three
dimensional space can be labelled by the radial distance,r, from the Z axis, the
angle, φ, the radius makes with the X-axis in the X-Y plane and the height, Z,
along the Z axis. This is depicted in Fig. ??. The values of these numbers are
confined to the ranges 0 ≤ r ≤ ∞, 0 ≤ φ < 2π and −∞ < Z < ∞ so that each
point has a unique triplet that labels it. The relation to the Cartesian coordinates
is obtained as
x = r cos φ,
y = r sin φ,
z = Z,
(2.20)
relations that can be easily inverted.
The unit vectors corresponding to each of these numbers point along the direction
of increasing coordinate at each point as shown in Fig. ??. These are easily related
to the Cartesian unit vectors as
r̂ = cos φx̂ + sin φŷ,
(2.21)
φ̂ = − sin φx̂ + cos φŷ,
(2.22)
Ẑ = ẑ.
(2.23)
It can be easily verified that the unit vectors are mutually perpendicular r̂ · φ̂ =
φ̂ · Ẑ = Ẑ · r̂ = 0 . It is clear from the above that the unit vectors change from
point to point, in this case they depend on the location through the angle φ. This
18
Mathematical essentials of Vectors and Vector Calculus
is unlike the unit vectors in the Cartesian system. Thus, we cannot thoughtlessly
move the unit vectors in or out across derivatives and integrals.
Another crucial difference comes from the consideration of the infinitesimal displacements along the three directions. Along the radial and axial directions, the
infinitesimal displacements corresponds to the change in the coordinates (dr and
dZ), which dimensions of length. Along the φ̂ direction, however, an infinitesimal
change in the coordinate (dφ) is an angle and translates to a length as rdφ (see
Fig. ??). Thus, there is a scale factor of r that depends on the given point in space.
Thus, the infinitesimal quantities in this coordinate system are:
1. The infinitesimal line element : d~r = r̂dr + φ̂rdφ + ẐdZ.
2. The infinitesimal volume element: d3 r = dr rdφ dZ.
3. The infinitesimal surface elements: d~sr = rdφ dZ r̂, d~sφ = dZ drφ̂, d~sZ =
dr rdφẐ.
2.3.3 Spherical coordinates
When a given problem has complete angular symmetry, i.e., when no direction is
preferrable over any other, the spherical coordinate system is very useful. Typically,
all properties of the system depend only on the absolute distance from a specific
point in space, which we will choose to be the origin. Now we will arbitrary choose
an axis, the z axis and a x axis on the plane perpendicular to the z axis and
containing the origin. Now any point in (three dimensional) space can be labelled
uniquely by a triplet of numbers: one representing the absolute radial distance (r)
to the origin, an angle θ indicating the angle between the radial line joining the
origin to the given point and the chosen z axis, and another angle φ that is the angle
between the projection of the radial line to the point on the X-Y plane and the
chosen x axis (see Fig. ??). The values of these numbers are confined to the ranges
0 ≤ r ≤ ∞, 0 ≤ theta ≤ π, and 0 ≤ φ < 2π so that each point corresponds to a
unique triplet that labels it. The relation to the Cartesian coordinates is obtained
as
x = r sin θ cos φ,
y = r sin θ sin φ,
z = r cos θ,
(2.24)
relations that can be easily inverted as
r = (x2 + y 2 + z 2 )1/2 ,
θ = cos−1
z
p
x2 + y 2 + z 2
!
,
φ = tan−1
y
x
. (2.25)
The unit vectors corresponding to each of these numbers point along the direction
of increasing coordinate at each point as shown in Fig. ??. These are easily related
2.3 Curvilinear geometries and coordinates
19
to the Cartesian unit vectors as
r̂ = sin θ cos φx̂ + sin θ sin φŷ + cos θẑ,
(2.26)
θ̂ = cos θ cos φx̂ + cos θ sin φŷ − sin θẑ,
(2.27)
φ̂ = − sin φx̂ + cos φŷ.
(2.28)
As in the case of the cylindrical system, these unit vectors point in different directions at different points. This is a general property of all curvilinear coordinate
systems which we will briefly discuss later. Hence one has to be careful while differentiating or integrating expressions containing these unit vectors.
The infinitesimal quantities in this system are:
1. The infinitesimal line element : d~r = r̂dr + θ̂rdθ + φ̂r sin θdφ.
2. The infinitesimal volume element: d3 r = dr rdθ r sin θdφ.
3. The infinitesimal surface elements: d~sr = rdθ r sin θdφ r̂, d~sθ = dr r sin θdφ θ̂,
d~sφ = dr rdθφ̂.
It can be seen that the scale factor r multiplies the infinitesimal change dθ to give
rise to an infinitesimal length rdθ along θ̂. Similarly, a scale factor r sin θ (projected
length of the radial vector on the X-Y plane) accompanies the infinitesimal quantity
dφ to give an infinitesimal length r sin θdφ along φ̂.
2.3.4 An orthogonal curvilinear coordinate system
We will not discuss, in detail, a curvilinear coordinate system, but will only list some
results that can be written down for orthogonal coordinate systems. For details, we
refer the reader to ?. Consider an invertible mapping to the coordinate system
(u1 , u2 , u3 ) from the Cartesian coordinates by the functions:
u1 = u1 (x, y, z),
u2 = u2 (x, y, z),
u3 = u3 (x, y, z).
(2.29)
It can be shown that the unit vectors are given by
ûi =
∇ui
.
|∇ui |
(2.30)
The infinitesimal displacement vector can be written as
d~r = h1 du1 uˆ1 + h2 du2 uˆ2 + h3 du3 uˆ3 ,
(2.31)
where the scale factors hi are given by
h2i = |∇ui |2 .
(2.32)
Now the infinitesimal volume is written as
d3 r = h1 h2 h3 du1 du2 du3 .
(2.33)
20
Mathematical essentials of Vectors and Vector Calculus
In general, we can also write down expressions for the gradient, divergence and
curl in the generalized coordinates using the scale factors
1 ∂f
1 ∂f
1 ∂f
+ uˆ2
+ uˆ3
,
h1 ∂u1
h2 ∂u2
h3 ∂u3
1
∂(A1 h2 h3 ) ∂(A2 h3 h1 ) ∂(A3 h1 h2 )
~=
∇·A
+
+
h1 h2 h3
∂u1
∂u2
∂u3
h1 û1 h2 û2 h3 û3 ∂
1
∂
∂
~=
∇×A
∂u1
∂u2
∂u3 h1 h2 h3 h A h A h A ∇f = uˆ1
1
1
2
2
3
(2.34)
(2.35)
(2.36)
3
It is easily seen that the scale factors for the cylindrical coordinates are given by
hr = 1,
hφ = r,
hZ = 1,
(2.37)
and for the spherical coordinates they are given by
hr = 1,
hθ = r,
hφ = r sin θ.
(2.38)
Knowledge of the scale factors enables us to carry out all the calculations on the
vectors fields in any desired coordinate system.
2.4 The Dirac δ− function
Consider the function
f (x) =
1
2w
0
∀ |x| < w
∀ |x| > w.
(2.39)
RL
It is evident that the integral −L f (x)dx = 1 if L > w. Now examine this function
in the limit w → 0. It is clear that the function is zero everywhere except the single
point x = 0 where it diverges, and yet the integral is exactly unity. This is contrary
to our usual understanding of (Riemann) integrals where the value of the integral
is zero unless the integration range is finite. In other words, a single point usually
has zero measure. Yet this mathematical object that results from a well defined
function in the limit w → 0 has a non-zero integral.
We will often face such mathematical objects in our study of electromagnetism.
Consider the following definition
0
∀ x 6= x0 ,
δ(x − x0 ) =
(2.40)
∞
∀ x = x0 ,
such that the integral
Z
b
δ(x − x0 )dx = 1,
(2.41)
a
if the interval [a, b] includes the singular point x0 and is zero otherwise. It is simple
2.4 The Dirac δ− function
21
to construct that δ(x − a) = δ(a − x). Note that the principal properties of this
object derive from the integral. The above mathematical construct was first formally
discussed in connection with quantum mechanics by a scientist called Dirac, and
it is are called the Dirac δ function. Although we call this object a function, it is
not a function in the conventional sense and belongs to a generalized class called
distributions by mathematicians.
The Dirac δ function can work as a sieve to pick out values of functions at specific
points. It is easily seen that
Z x0 +
Z b
δ(x − x0 )f (x)dx = f (x0 ),
(2.42)
δ(x − x0 )f (x)dx = lim
→0
a
x0 −
where f (x) is a usual continuous function. Sometimes it is convenient to work with
certain functions that become a δ function in limiting cases. The rectangular function presented above is one example. Other possible examples include a Gaussian
function
(x − x0 )2
1
,
(2.43)
exp −
δ(x − x0 ) = lim √
σ→0
2σ 2
2πσ
and a Lorentzian function
δ(x − x0 ) = lim
σ→0
1
σ
.
π (x − x0 )2 + σ 2
(2.44)
In both of the cases above, σ is linearly proportional to the full width of the functions
where the value of the singly peaked functions falls to half their peak value. As this
width falls to zero in the limit, the peak value rises keeping the value of the integral
constant (unity). The δ function can also be interpreted as the derivative of a stepfunction at the point of discontinuity.
Consider the Heaviside step function
1
∀ x > x0 ,
Θ(x − x0 ) =
(2.45)
0
∀ x < x0 ,
its derivative can be shown to be a δ function. We can do this by showing that it
has the property of the δ function. The derivative is clearly zero everywhere except
at x0 , and for an arbitrary function that is continuous at x0 , the integral
Z b
Z b
df
d
b
θ(x − x0 ) dx
f (x) Θ(x − x0 )dx = [f (x)Θ(x − x0 )]a −
dx
dx
a
a
Z b
df
= f (b) −
dx
x0 dx
= f (b) − [f (b) − f (x0 )] = f (x0 )
(2.46)
where the interval [a, b] is assumed to contain the point x0 , and we have integrated
by parts. Clearly the derivative of the step-function has all the essential properties
of the δ function.
22
Mathematical essentials of Vectors and Vector Calculus
The idea of the δ function as a point of singularity but with a finite integral
is easily extended to higher dimensions. In three dimensional space, we have the
integral
Z
δ(~r − r~0 )d3 r = 1
(2.47)
V
if the integration volume V contains the point r~0 and is zero otherwise. In Cartesian
coordinates, it is straightforward to represent the δ function as a product of onedimensional δ functions,
δ(~r − r~0 ) = δ(x − x0 ) δ(y − y0 ) δ(z − z0 ).
(2.48)
Representation of higher dimensional δ functions in other co-ordinate systems will
be discussed in the next section. Note that the one dimensional δ function has
dimensions of inverse length, Hence the three-dimensional δ function has dimensions of inverse volume. This gives rise to the interpretation that the δ function is
effectively a density.
In general curvilinear coordinate systems, the infinitesimal volumes depend on
the point and it becomes important to normalize the δ function to account for this
change in the density. For a δ function located on the point (u01 , u02 , u03 ), we write
δ(~r − r~0 ) =
1
δ(u1 − u01 )δ(u2 − u02 )δ(u3 − u03 ).
h1 h2 h3
(2.49)
Unless properly normalized, the δ function would begin to have different weights
depending on where it is placed. Overall the δ function should be defined such that
the integral over a volume containing the point where the singularity is located
should yield unity. Thus, in spherical coordinates the δ function would be written
as
1
δ(r − r0 )δ(θ − θ0 )δ(φ − φ0 ).
(2.50)
δ(~r − r~0 ) = 2
r sin θ
Special mention must be made of points of singularity such as the origin or points
on the z axis where the spherical coordinates θ and φ may become ill-defined, i.e.,
the point is multiply described by the curvilinear coordinates. In such cases, if the
coordinate u3 multiply describes the point where the δ function is located, there
will be no such factor such as δ(u3 − u03 ) in the representation for the δ function,
since the value of u03 would be non-unique and ill-defined. Hence the representation
the curvilinear coordinate system would only appear as
δ(~r − r~0 ) =
h1 h2
EXAMPLES:
1
Rb
a
h3 du3
δ(u1 − u01 )δ(u2 − u02 )
(2.51)
2.4 The Dirac δ− function
23
1. Consider a point charge q located at the origin. In cylindrical coordinates, the
corresponding charge density would be described as
1
δ(r)δ(z).
2πr
In spherical coordinates, the representation would be
δ(~r) = q
1
δ(r).
4πr2
2. Consider a charged thin disk of radius R on the carrying a charge per unit area
of σ lying on the X − Y plane. The volume charge density can be represented
in cylindrical coordinates as
δ(~r) = q
ρ(~r) = σδ(z)Θ(R − r),
while the representation in spherical coordinates is
1
ρ(~r) = σ δ(θ − π/2)Θ(R − r).
r
Note that the Heaviside step function has been used to confine the charge toa
radius smaller than R.
3. Consider a line charge with linear charge density λ per unit length, located along
the Z axis. In Cartesian coordinates, this is is easily represented as
ρ(~r) = λδ(x)δ(y),
while in the cylindrical coordinates, we can write
1
δ(r),
2πr
and in the spherical coordinates, the representation would be
ρ(~r) = λ
ρ(~r) = λ
1
[δ(θ) + δ(θ − π)].
2πr2 sin θ
3
Static charges, electric field and electric potential
3.1 Concept of the electric field
The Coulomb’s law gives the force exerted by one charge (q1 ) on another charge
(q2 ) as
F~ =
1 q1 q2 (~r2 − ~r1 )
4πε0 |~r2 − ~r1 |3
(3.1)
where ~r1 is the position vector of the location of the charge q1 and ~r2 is the position
vector location of the charge q2 . This is essentially action at a distance, whereby the
force due to the first charge is instantaneously felt by the second charge. However,
we know from the theory of special relativity, that if one of the charges is moved
suddenly, the information of the changed location and hence the changed force
cannot be felt immediately. This information can only be known after a minimum
time of r21 /c which is the time that it takes light to travel from one point to another.
Of course, we could argue that when we are discussing electrostatics, one cannot
talk of time dependent dynamic phenomena. However, this point makes us realize
that if we took the view that the forces between the charges are instantaneous, then
this would need to be changed once we began to talk to time-dependent phenomena.
There is an alternative and more fruitful viewpoint to take in this respect. We
can think of a charge affecting the space around itself such that any other charge
feels a force when placed in the space influenced by it. This property of the space
around the point charge is called the Electric field. It is defined as the force felt by
a unit charge. Hence, from Coulomb’s law, we can write that the electric field at a
point ~r due to a point charge located at r~0 is
~ r) =
E(~
1 q1 (~r − r~0 )
.
4πε0 |~r − r~0 |3
(3.2)
~ r) in a
This is called the Coulomb field of a point charge. Once the electric field E(~
given region is known, the force felt by a (test) charge in the given region is given
by
~ r).
F~ = qtest E(~
(3.3)
3.1 Concept of the electric field
25
It is assumed that the placement of the test charge in that region does not affect
the charge configuration that gives rise to the electric field. For example, it should
not cause the first charge to move - we have assumed that the first charge is fixed
at the location r~0 in the above equations. If the test charge is small enough, then
such assumptions are usually valid. .
3.1.1 Superposition of electric fields and Charge distributions
Since the Coulomb forces due to various charges superpose (this is an experimental
fact), it is straightforwardly seen that one has a linear superposition of the electric
~ 1 (~r) is the electric field at the location ~r due to a point charge q1 ,
fields too. If E
~
E2 (~r) is the electric field due to a point charge q2 , and so on, then the net electric
field at ~r is given by
~ r) = E
~ 1 (~r) + E
~ 2 (~r) + E
~ 3 (~r) + · · · ,
E(~
(3.4)
i.e. it is the vectorial sum of the electric fields produced by the individual charges.
Taking the location of the point charge qi to be ~ri , we can use the Coulomb law
in the superposition above and write for the net electric field from a set of n point
charges as
n
X
1 qi (~r − ~ri )
~ r) =
.
(3.5)
E(~
4πε
r − ~ri |3
0 |~
i=1
Next, we ask whether a point charge is actually found in nature or is it a mere
idealization like a point mass? We believe that electrons (as well as positrons and
other leptons) are truly point charges – this is because we have never found any
internal structure to the electron in any of our experiments1 . However, in most of
our everyday experiments and devices, many thousands, if not millions and billions
or more, electrons are involved. Further, we never try to measure the given charge
at a given point. Usually the question we ask is, how much charge is present within
a given volume? The volume depends on the resolution of the measurement we
perform – the although we can improve the spatial resolution by better measurement
methods and techniques, the measurement volume is never really zero and is in
fact large compared to the atomic volumes in most cases. Thus, it makes better
sense to talk of average charge densities at a slightly more coarse-grained level,
where the graininess of the electron and the electronic charge is not really visible.
For example, if we consider the volume of a typical capacitor, it is several cubic
millimetres and the charge density in the volume concerned can be considered to
almost vary continuously. This is much like a fluid where the graininess of the
1
These experiments involve colliding electrons against electrons at very high energies and observing
the energies of the scattered electrons and other particles that are produced in the collision. The
internal structure, if any, can be deduced from the variation of these energies and particles with
direction.
26
Static charges, electric field and electric potential
atomic structure making up the fluid is averaged out due to the presence of a very
large number of atoms involved.
In the above spirit, we will now consider continuous distributions of charges where
the amount of charge in a small (infinitesimal) volume around a given point (given
by the position vector ~r) is defined as
dq = ρ(~r)dr
(3.6)
where ρ(~r) is a smooth continuous function describing the charge density in space.
It is apparent from our above definition that the charge density of a point charge
would be singular. From our discussion of the Dirac δ functions in in Chapter 2, it
is clear that the charge distribution of a point charge is a δ function at the given
location (r~0 ) weighted by the magnitude of the charge: qδ(~r − r~0 . In most other
physical macroscopic charge distributions, we have a continuous charge density that
avoids all such problems.
Now we can generalize our expression for the electric field for a set of discrete
charges to a continuous (but finite) charge distribution as an integral over the charge
density
Z
ρ(~r0 )(~r − ~r0 ) 3 0
1
~
d r.
(3.7)
E(~r) =
4πε0
|~r − ~r0 |3
Note that the integral has to be carried out over the volume where the charges
are present. This integral can be carried out for a few situations analytically. But
in most cases, the integrals become extremely cumbersome to evaluate and would
mostly have to be carried out by numerical or alternative methods that we will
learn a bit later. Further, note that in many experiments, specifying the charge
density is quite difficult as we will learn in due process.
EXAMPLES Consider a uniformly charged thin disk of radius a and total charge
q. We will find the electric field at points on the axis of the disk. We will take
the z axis to lie along the axis of the disk that is taken to be on the X-Y plane.
The charge density of the disk can be written as ρ(~r) = q/(πa2 )δ(z)Θ(a − r) in
cylindrical coordinates. The electric field at points on the Z− axis then comes out
to be (see Figure (??),
Z Z Z
q/(πa2 )δ(z)Θ(a − r)(z ẑ − r0 r̂) 0
1
~
r dzdr0 dφ,
R(z)
=
4πε0
(r02 + z 2 )3/2
Z a
r0 dr0
q
2π
ẑz
=
02
2 3/2
4π 2 a2 ε0
0 (r + z )
q
z
ẑ 1 − 2
=
(3.8)
2πa2 ε0
(z + a2 )1/2
where
R 2π we have written out the integral in cylindrical coordinates and used that
r̂dφ = 0.
0
3.2 Gauss’s law of Electrostatics
3.2 Gauss’s law of Electrostatics
3.3 The curl of the electric field
3.3.1 The electrostatic Potential
3.4 Energy associated with charge distributions
3.4.1 Problems regarding energy of a point charge
3.5 Conductors and capacitance
3.5.1 A cavity within a conductor
3.5.2 Surface charge on a conductor
3.5.3 Surface force on a conductor
3.5.4 Energy of a capacitor
27
4
Calculating the Electric field and potential
4.1 Poisson and Laplace equations: Boundary Value problems
4.2 Boundary conditions on the electrostatic field across charged
surfaces
4.3 Uniqueness theorems
4.4 The method of images
4.4.1 A conducting infinite plane
4.4.2 A conducting sphere
4.5 Sorting out some important problems
4.5.1 A conducting sphere placed in a homogeneous electric field
4.5.2 Screening: fields within a atom
4.5.3 Why is a colloid suspension stable ?
4.5.4 Oscillations of a plasma
5
Approximate description of the electric field at
far-off points
5.1 Multipole expansions
5.2 Fields and potential of dipoles
5.3 Force, torque and energy associated with a dipole in electrostatic
fields
6
Electrostatics of material media
6.1 Ideas of homogenization of the electric field in a material medium
6.2 Bound charges, polarizability and macroscopic polarization
6.3 Electric field inside a material medium and The displacement field
6.4 Dielectric susceptibility and dielectric permittivity
6.5 The electrostatic energy inside a dielectric medium
6.6 Sorting out some important problems
6.6.1 The electric fields of a uniformly polarized dielectric sphere
6.6.2 A dielectric sphere placed in a homogeneous electric field
6.6.3 Making a composite material with high dielectric permittivity
6.6.4 Fields of a point charge placed near a flat, large dielectric
medium
6.6.5 Placing a dielectric inside a capacitor
7
Magnetic fields
7.1 Biot–Savart law for the magnetic field
7.1.1 Experimental verification of Biot–Savart law
7.2 Calculating fields for simple current configurations
7.3 Current distributions and magnetic fields for distributed currents
7.4 Forces between current carrying conductors
7.5 Divergence and curl of the magnetic field and Amperés Law
7.6 The magnetic vector potential
7.7 Boundary conditions on magnetic fields across current sheets
8
Forces on a charge in electric and magnetic fields:
The Lorentz force
8.1 The Cyclotron frequency
8.2 Motion in parallel electric and magnetic fields
8.3 Motion in crossed electric and magnetic fields
8.3.1 Confining and focussing charges
9
Magnetostatics of material media
9.1 Multipole expansions for the magnetic field
9.2 Magnetic field and magnetic vector potential of a magnetic dipole
~ field in a material medium
9.3 Magnetic fields and the H
9.4 Basic ideas of para-, dia-, ferro- magnetic media
9.5 Magnetic fields of magnetized objects
Notes
Author index
Subject index