Notes
Electromagnetic Theory – I (PHYS 3117)
Chapter: 1 Electrostatics
Electrostatics: The branch of Physics which deals with the charges at rest.
Electric Charge: Charge is a fundamental property of the elementary particles which makes
up matter. Namely there are two types of charges positive and negative.
Smallest charge on a particle is 1.6 × 10-19 C (charge on electron).
Like charges repel each other and unlike charges attract each other. In a closed system the net
charge is conserved. Each charge possess electric field around it, whose strength depends upon
charge.
Coulomb’s Law:
It was observed quite early that when particles carrying electric charge are brought closer, they
experience a force, and this force was called the electric force. The main question that one studies in
electrostatics is the analysis of the electric force experienced by a given charge due to a complicated
distribution of static electric charges in space. (As a side remark, let us simply note here that although
at the moment it may seem like the course is only concerned with developing techniques for solving
problems in electromagnetism, the techniques are quite general and are so powerful that they may be
used in any other field of research as well. In that sense, we can think of the material of the course as
developing powerful techniques for solving theoretical problems through examples of
electromagnetism.) Fundamental to this study, therefore, is the understanding of the force between a
pair of static charges separated by a given distance. This question was, in fact, studied by Coulomb in
a series of impressive experiments and he found that the electric force between a pair of static particles
carrying electric charge
1. is linearly proportional to the individual charges on each of the particles,
2. varies inversely as the square of the distance between the two particles,
3. and is a vector along the line joining the positions of the two particles carrying charge. It is
attractive if the charges of the two particles have opposite sign and is repulsive otherwise.
Quantitatively, we can, therefore, write that the force experienced by a particle with charge q at the
coordinate r because of the presence of a static particle carrying charge q1 at the coordinate r1 is
given by
𝑭(𝑟 − 𝑟1 ) = 𝑘
𝑞𝑞1
𝑞𝑞1
𝑞𝑞1
(𝒓 − 𝒓1 ) = 𝑘 3 𝑅 = 𝑘 2 𝑅
3
|𝑟 − 𝑟1 |
|𝑅|
|𝑅|
Here, we have defined R = r - r1 (see Fig. 1.1) and have used the notation (which we will use throughout
the lectures) that a boldface quantity represents a vector while a boldface quantity with a “hat” simply
stands for a unit vector and, in the above equation, k represents the constant of proportionality.
Figure 1.1: Direction of the electrostatic force experienced by a charge q at r due to a charge q1 of the
same sign at coordinate r1.
There are several things to note about the force from the expression in (1.1). First, it is translationally
invariant, namely, under a translation of the coordinate system, the expression is unchanged.
Sometimes, one takes advantage of this to translate the coordinate system such that q1 is at the origin,
in which case, the force has the simple form
𝑞𝑞
𝑞𝑞
𝑭(𝑟 − 𝑟1 ) = 𝑘 |𝒓|31 𝒓 = 𝑘 |𝒓|21 𝑟
1.2
Where r denotes the coordinate of the charge q in the new frame (with respect to the new origin).
Second, the value of the constant k depends on the system of units used. In the CGS (Gaussian) units
(which we will work with throughout these lectures), where distances are measured in centimetres and
the force in dynes, k = 1 while in the MKS (SI) units (which is used mostly in engineering applications),
where distances are measured in meters and the force in Newton’s, k = 10-7 N-m2. Here “N” stands for
Newton, “C” for Coulomb and “m” for meter and c is the speed of light (just the value ≈ 3×108 in the
MKS system without the dimensions). The rationale behind the choice of different units is as follows.
In applications (SI units), currents are measured in units of Ampere which is defined to be Coulomb/sec.
Therefore, one can define the unit of electric charge, Coulomb, from Ampere’s law which involves the
magnetic force between two currents. Once the unit of charge, Coulomb, is defined in this way,
consistency determines the constant k in the Coulomb’s law. (Basically, once the unit of charge has
been defined from the magnetic force, it has to be consistent with the definition of force in Coulomb’s
law. The peculiar value of the constant arises because it involves the vacuum magnetic permeability
since it originates from the definition of the magnetic force and the constant can be identified with k =
1/4πεo where εo is the dielectric permittivity of the vacuum through usual relations.) On the other hand,
in the CGS (Gaussian) units, the desire is more to be consistent with the requirements of relativistic
invariance. We know from the studies in relativity that electric and magnetic fields can be mapped to
each other under a Lorentz transformation. The simplest way to see this is to note that if we have a static
charge, it only produces an electric field. However, in a different Lorentz frame, the charge would be
moving giving rise to a current and, therefore, to a magnetic field as well. Thus, relativistic invariance
treats electric and magnetic fields on the same footing. From the form of Lorentz transformations, we
can also see that they have to have the same dimension. The choice k = 1 enforces this and we see that
if we choose k = 1, then, Coulomb’s law can be thought of as defining the unit of electric charge in the
CGS units, which is an “esu”. (The electric force between two charges of magnitude 1 esu each, at a
separation of 1 cm, is defined to be 1 dyne.) It is quite easy to see from this that (remember that 1N =
105 dynes)
1 esu = 1 stat Coulomb = (10 c)-1 C = (3 × 109)-1 C.
In fact, let us suppose, 1 esu = α C. Then,
(1.3)