Contents
Syllabus 2
1 Electric Field and Electric Potential 4
1.1 Review of Vector Mathematics* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Introduction to Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Vector Calculus* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Gauss’ Law and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5 Electric (or Electrostatic) Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.6 Advanced Methods in Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2 Electrostatics in Conductors and Dielectrics 50
2.1 Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2 The Electric Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Polarization in Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4 Capacitance and Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 The Magnetic Field 71
3.1 Force due to the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 The Biot-Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4 Vector Properties of the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Magnetic Properties of Matter 94
4.1 The Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Magnetization in Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Linear Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Electromagnetic Induction 103
5.1 Electromotive Force and Magnetic Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4 Energy Stored in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Electromagnetic Wave Theory 117
6.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3 Polarization of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Appendix: References and Suggested Readings 130
2
, 24PHYMAJ103 - Electricity and Magnetism
Syllabus
Approx.
Units Content Lecture
Hours
Electric Field and Electric Potential: Electric field: Electric
field lines. Electric flux. Coulomb’s law. Gauss’ Law with
applications to charge distributions with spherical, cylindrical
Unit I and planar symmetry. Conservative nature of Electrostatic Field. 12
Electrostatic Potential. Electrostatic boundary conditions.
Laplace’s and Poisson equations. Uniqueness theorem. Method
of images.
Electrostatics in Conductors and Dielectrics: Conductors and
Dielectrics, Electric Dipole, Dielectric Polarization, Gauss’s law
in Dielectric medium, Boundary Conditions.
Unit II 10
Electrostatic energy of a system of charged conductors,
capacitors and their combinations, energy stored in a charged
capacitor.
Magnetic Field: Magnetic force between current elements and
definition of Magnetic Field B. Biot-Savart’s Law and its simple
applications: straight wire and circular loop. Current Loop as a
Magnetic Dipole and its Dipole Moment. Ampere’s Circuital
Law and its application to (1) infinite straight wire, (2) Infinite
Unit III 8
planar surface current and (3) Solenoid. Properties of B: curl and
divergence. Axial vector property of B and its consequences.
Magnetic Vector Potential. Magnetic Force on (1) point charge
(2) current carrying wire (3) between current elements. Torque
on a current loop in a uniform Magnetic Field.
Magnetic Properties of Matter: Magnetization vector (M).
Magnetic Intensity (H). Magnetic Susceptibility and
Unit IV 5
permeability. Relation between B, H, M. Comparison of Dia-,
Para- and Ferromagnetism. B-H curve and hysteresis loss.
Electromagnetic Induction: Faraday's law. Self-Inductance and
Mutual Inductance. Neumann’s Formula. Energy stored in a
Magnetic Field.
Electromagnetic Theory: Maxwell’s equations, Poynting’s
Unit V 10
vector and Poynting’s theorem. Electromagnetic wave
propagation (qualitative treatment). Polarization of plane
electromagnetic waves (qualitative treatment), Methods of
producing plane polarized light: simple reflection, double
refraction and dichroism, Malus law.
, CHAPTER 1
Electric Field and Electric Potential
1.1. Review of Vector Mathematics*
A vector is a quantity that gives us two pieces of information: how strong something that we are
measuring is (magnitude), and what direction it is acting towards. For example, let’s say that I’m
walking from Don Bosco Block to Nazareth Block at a fast walk pace. This is sort of a vector quantity
in the sense that you know how much (fast walk pace) and which direction (DB Block to NZ Block),
but it’s too vague to be useful in actual calculations. However if I say I’m walking at a speed of 2
m/s from Don Bosco Block in the North Direction, then this is a vector as it gives us a mathematically
workable quantity and a specific direction. You will note that the magnitude itself is just a number
(with a unit attached), the direction information has to be given separately.
Since we will be using vectors as a measure of some physical quantity, its magnitude will usually
have a specific unit (m/s, in my case). Additionally, we usually don’t use cardinal directions in our
problems. Thus, to specify directions, we have to set up a coordinate system. Depending upon the
coordinate system we use, we can describe the vector in Cartesian coordinates (î, ĵ, k̂), spherical
coordinates (r̂, θ̂, ϕ̂) etc. Since we have the freedom to choose our coordinate system, the same vector
can have very different mathematical descriptions in different coordinate systems.
⃗ . Depending upon the coordinate system used, we can describe it using
Let us define some vector a
components that use the axes of the coordinate system to specify the vector. In Cartesian coordinates,
we can describe it as
⃗ = ax î + ay ĵ + az k̂
a (1.1.1)
In spherical coordinates, we have
⃗ = ar r̂ + aϕ ϕ̂ + aθ θ̂
a (1.1.2)
and in cylindrical coordinates, we have
⃗ = ar r̂ + aϕ ϕ̂ + az ẑ
a (1.1.3)
Please note that in each of these three cases, the vector components (ax , ar etc.) are not vectors. They
are to be treated as ordinary numbers (scalars). Combining them with the axis vector (î, θ̂ etc.,also
formally called the basis vector) gives it the requisite direction.
4
, CHAPTER 1. ELECTRIC FIELD AND ELECTRIC POTENTIAL 5
Note: In cylindrical coordinates, the standard symbol for the radial term is ρ instead of r (that I’m
using above). However ρ is also the standard symbol representation of charge density in electrostat-
ics, so I’m sticking to using r in cylindrical coordinates to prevent confusion. If you are following
Griffith’s textbook, you’ll notice that he uses s instead.
For each coordinate system, the magnitude of the vector is then described as
q
Cartesian: a = |⃗
a| = a2x + a2y + a2z (1.1.4)
Spherical: a = |⃗
a| = ar (1.1.5)
q
Cylindrical: a = |⃗
a| = a2r + a2z (1.1.6)
A unit vector is defined as a dimensionless vector with a magnitude of 1. Unit vectors are very useful
in defining directions, and are usually denoted with a special "hat" symbol to indicate their unique
nature. Our axis/basis vectors (î, θ̂ etc.) are all unit vectors. To find a unit vector that is pointing
along the same direction as the vector a ⃗ , we can simply divide the vector by its magnitude
⃗
a ⃗
a
â = = (1.1.7)
a |⃗
a|
Using equations 1.1.4, 1.1.5 or 1.1.6, we can use equation 1.1.7 above to convert an ordinary vector
into the required unit vector.
1.1.1 Components of a vector
Components in 2D
Let us look at finding the components of a vec-
tor. Figure1 1.1.1 shows a 2D vector displayed
in a Cartesian coordinate system. Let us say ⃗F
is describing a force with a magnitude of 5 N at
a direction roughly 37◦ above the X-axis. Using
equation 1.1.1, we have
⃗F = Fx î + Fy ĵ + Fz k̂
Since there is no Z-axis in this problem, we can
simply ignore the k̂ term (or set Fz = 0). We
have the magnitude and direction, the question
is how to find the components. From a little bit F IGURE 1.1.1: Components of a 2D vector
of trigonometry, we can see that Fx /F = cos θ
and Fy /F = sin θ . So we can write
Fx = F cos θ = 5 × cos 37◦ = 4 N
Fy = F sin θ = 5 × sin 37◦ = 3 N
which gives us a final expression of
⃗F = 4 î + 3 ĵ
1
Image taken from Boas, M.L. (2006). Mathematical methods in the physical sciences (3rd ed.). John Wiley & Sons Inc. Page
97.
Syllabus 2
1 Electric Field and Electric Potential 4
1.1 Review of Vector Mathematics* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Introduction to Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Vector Calculus* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Gauss’ Law and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5 Electric (or Electrostatic) Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.6 Advanced Methods in Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2 Electrostatics in Conductors and Dielectrics 50
2.1 Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2 The Electric Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Polarization in Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4 Capacitance and Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 The Magnetic Field 71
3.1 Force due to the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 The Biot-Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4 Vector Properties of the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Magnetic Properties of Matter 94
4.1 The Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Magnetization in Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Linear Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Electromagnetic Induction 103
5.1 Electromotive Force and Magnetic Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4 Energy Stored in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Electromagnetic Wave Theory 117
6.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3 Polarization of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Appendix: References and Suggested Readings 130
2
, 24PHYMAJ103 - Electricity and Magnetism
Syllabus
Approx.
Units Content Lecture
Hours
Electric Field and Electric Potential: Electric field: Electric
field lines. Electric flux. Coulomb’s law. Gauss’ Law with
applications to charge distributions with spherical, cylindrical
Unit I and planar symmetry. Conservative nature of Electrostatic Field. 12
Electrostatic Potential. Electrostatic boundary conditions.
Laplace’s and Poisson equations. Uniqueness theorem. Method
of images.
Electrostatics in Conductors and Dielectrics: Conductors and
Dielectrics, Electric Dipole, Dielectric Polarization, Gauss’s law
in Dielectric medium, Boundary Conditions.
Unit II 10
Electrostatic energy of a system of charged conductors,
capacitors and their combinations, energy stored in a charged
capacitor.
Magnetic Field: Magnetic force between current elements and
definition of Magnetic Field B. Biot-Savart’s Law and its simple
applications: straight wire and circular loop. Current Loop as a
Magnetic Dipole and its Dipole Moment. Ampere’s Circuital
Law and its application to (1) infinite straight wire, (2) Infinite
Unit III 8
planar surface current and (3) Solenoid. Properties of B: curl and
divergence. Axial vector property of B and its consequences.
Magnetic Vector Potential. Magnetic Force on (1) point charge
(2) current carrying wire (3) between current elements. Torque
on a current loop in a uniform Magnetic Field.
Magnetic Properties of Matter: Magnetization vector (M).
Magnetic Intensity (H). Magnetic Susceptibility and
Unit IV 5
permeability. Relation between B, H, M. Comparison of Dia-,
Para- and Ferromagnetism. B-H curve and hysteresis loss.
Electromagnetic Induction: Faraday's law. Self-Inductance and
Mutual Inductance. Neumann’s Formula. Energy stored in a
Magnetic Field.
Electromagnetic Theory: Maxwell’s equations, Poynting’s
Unit V 10
vector and Poynting’s theorem. Electromagnetic wave
propagation (qualitative treatment). Polarization of plane
electromagnetic waves (qualitative treatment), Methods of
producing plane polarized light: simple reflection, double
refraction and dichroism, Malus law.
, CHAPTER 1
Electric Field and Electric Potential
1.1. Review of Vector Mathematics*
A vector is a quantity that gives us two pieces of information: how strong something that we are
measuring is (magnitude), and what direction it is acting towards. For example, let’s say that I’m
walking from Don Bosco Block to Nazareth Block at a fast walk pace. This is sort of a vector quantity
in the sense that you know how much (fast walk pace) and which direction (DB Block to NZ Block),
but it’s too vague to be useful in actual calculations. However if I say I’m walking at a speed of 2
m/s from Don Bosco Block in the North Direction, then this is a vector as it gives us a mathematically
workable quantity and a specific direction. You will note that the magnitude itself is just a number
(with a unit attached), the direction information has to be given separately.
Since we will be using vectors as a measure of some physical quantity, its magnitude will usually
have a specific unit (m/s, in my case). Additionally, we usually don’t use cardinal directions in our
problems. Thus, to specify directions, we have to set up a coordinate system. Depending upon the
coordinate system we use, we can describe the vector in Cartesian coordinates (î, ĵ, k̂), spherical
coordinates (r̂, θ̂, ϕ̂) etc. Since we have the freedom to choose our coordinate system, the same vector
can have very different mathematical descriptions in different coordinate systems.
⃗ . Depending upon the coordinate system used, we can describe it using
Let us define some vector a
components that use the axes of the coordinate system to specify the vector. In Cartesian coordinates,
we can describe it as
⃗ = ax î + ay ĵ + az k̂
a (1.1.1)
In spherical coordinates, we have
⃗ = ar r̂ + aϕ ϕ̂ + aθ θ̂
a (1.1.2)
and in cylindrical coordinates, we have
⃗ = ar r̂ + aϕ ϕ̂ + az ẑ
a (1.1.3)
Please note that in each of these three cases, the vector components (ax , ar etc.) are not vectors. They
are to be treated as ordinary numbers (scalars). Combining them with the axis vector (î, θ̂ etc.,also
formally called the basis vector) gives it the requisite direction.
4
, CHAPTER 1. ELECTRIC FIELD AND ELECTRIC POTENTIAL 5
Note: In cylindrical coordinates, the standard symbol for the radial term is ρ instead of r (that I’m
using above). However ρ is also the standard symbol representation of charge density in electrostat-
ics, so I’m sticking to using r in cylindrical coordinates to prevent confusion. If you are following
Griffith’s textbook, you’ll notice that he uses s instead.
For each coordinate system, the magnitude of the vector is then described as
q
Cartesian: a = |⃗
a| = a2x + a2y + a2z (1.1.4)
Spherical: a = |⃗
a| = ar (1.1.5)
q
Cylindrical: a = |⃗
a| = a2r + a2z (1.1.6)
A unit vector is defined as a dimensionless vector with a magnitude of 1. Unit vectors are very useful
in defining directions, and are usually denoted with a special "hat" symbol to indicate their unique
nature. Our axis/basis vectors (î, θ̂ etc.) are all unit vectors. To find a unit vector that is pointing
along the same direction as the vector a ⃗ , we can simply divide the vector by its magnitude
⃗
a ⃗
a
â = = (1.1.7)
a |⃗
a|
Using equations 1.1.4, 1.1.5 or 1.1.6, we can use equation 1.1.7 above to convert an ordinary vector
into the required unit vector.
1.1.1 Components of a vector
Components in 2D
Let us look at finding the components of a vec-
tor. Figure1 1.1.1 shows a 2D vector displayed
in a Cartesian coordinate system. Let us say ⃗F
is describing a force with a magnitude of 5 N at
a direction roughly 37◦ above the X-axis. Using
equation 1.1.1, we have
⃗F = Fx î + Fy ĵ + Fz k̂
Since there is no Z-axis in this problem, we can
simply ignore the k̂ term (or set Fz = 0). We
have the magnitude and direction, the question
is how to find the components. From a little bit F IGURE 1.1.1: Components of a 2D vector
of trigonometry, we can see that Fx /F = cos θ
and Fy /F = sin θ . So we can write
Fx = F cos θ = 5 × cos 37◦ = 4 N
Fy = F sin θ = 5 × sin 37◦ = 3 N
which gives us a final expression of
⃗F = 4 î + 3 ĵ
1
Image taken from Boas, M.L. (2006). Mathematical methods in the physical sciences (3rd ed.). John Wiley & Sons Inc. Page
97.
