Electromagnetism/An introduction to the subject

Electromagnetism is the study of interactions among charges and description of electromagnetic fields. Electromagnetic interaction is one of four fundamental interactions of nature. The other three interactions are gravitational interaction, strong interaction and weak interaction. The theory of electromagnetism is a vast theory. The branch was developed over a long period. Earlier, the phenomena of electricity and magnetism were understood differently. Oersted's experiment first revealed that electric currents interacted with magnets. Later, Maxwell's theory of electromagnetic fields unified the study of optics with electromagnetism as light was discovered to be electromagnetic waves. We experience electromagnetic interaction everyday. It is the electromagnetic interaction that builds an atom or a molecule. Life on earth is an example of electromagnetic interaction. And electromagnetic radiation is all around us. We experience the electromagnetic interaction most among the four fundamental interactions.

Electric charge
It is very difficult to say exactly what is an electric charge. But we know that electric charges exist. Electric charges create electromagnetic fields and electromagnetic fields have effects on electric charges. How electromagnetic fields force on charge is given by Lorentz force law and with this we can measure the magnitude of a charge.

Distribution of charge
Electric charge can be distributed in the space both discretely and continuously. In discrete distribution, charges can be considered as point sources. In continuous distribution, we assume that the configuration is a result of differential charges in every point of the system. The differential amount of charge is
 * $$ dq $$ ～ $$ \lambda dl $$ ～ $$ \sigma da $$ ～ $$ \rho d \tau $$

where $$ \lambda $$ is the line charge density, $$ \sigma $$ is the surface charge density and $$ \rho $$ is the volume charge density.

Current
A flow of charges is called a current. The current is defined as the amount of charge flowing through any cross sectional area per unit time. It the current is $$ I $$ then the amount of charge passed through that volume in a time interval from $$ t=t_1 $$ to $$ t=t_2 $$ is
 * $$ Q= \int_{t_1}^{t_2} I(t) dt $$

When the current is on a line with the line charge density $$ \lambda $$ and velocity of charge is $$ \mathbf{v} $$ then the current is
 * $$ \mathbf{I}= \lambda \mathbf{v} $$

When the current is on surface, and the surface charge density is $$ \sigma $$ then the surface current density $$ \mathbf{K} $$ is defined as
 * $$ \mathbf{K}= \sigma \mathbf{v} $$

An alternative definition is
 * $$ dI= \mathbf {K} \cdot d \mathbf{l} $$

with $$ dI $$ is infinitesimal current through the infinitesimal line element $$ d \mathbf{l} $$.

When the current is a volume current, the charge density is $$ \rho $$ and velocity of charges is $$ \mathbf{v} $$ then the volume current density $$ \mathbf{J} $$ is defined as
 * $$ \mathbf{J} = \rho \mathbf{v} $$

An alternative definition is
 * $$ dI= \mathbf {J} \cdot d \mathbf{a} $$

with $$ dI $$ is infinitesimal current through the infinitesimal surface element $$ d \mathbf{a} $$.

Positive and negative charge
There are two kinds of electric charge. We call one kind positive and the other kind negative. It was our choice to call one kind of them as positive and the other kind negative. But such formulation is very useful to write down the mathematical formulae. An electron is negatively charged and a proton is positively charged.

Conservation of charge
Electric charge is a conserved quantity. It can not be created and destroyed (We can place one positive and one negative charge close enough so that it may look like no charge. But this does not destroy the charges. Also we can not create charges out of nothing). This statement is called the statement of global conservation of charge. But electric charges are also locally conserved i.e. in a system, the number and amount of charge is constant so far as no charge enters in the system and crosses the boundary of the system. This can be stated mathematically as
 * $$ \oint\limits_{S} \mathbf{J} \cdot d \mathbf{a} + \frac {d Q}{d t} =0 $$

Here $$ Q $$ is the amount of charge within the surface $$ S $$ and the direction of current is outwards of the surface.

Quantization of charge
Charge is a quantized quantity. It comes in integer multiple of a certain quantity of charge of an electron, denoted by $$ e $$ (Although electron is negatively charged, we use $$ e $$ to denote an amount of positive charge equal to the magnitude of charge of an electron or a proton). We can have $$ \pm e $$, $$ \pm 2e $$ charges but never $$ \frac {e}{2} $$ charge. (According to the standard model, quarks have $$ \pm \frac {e}{3} $$ and $$ \pm \frac {2e}{3} $$ charges. But these are individually very unstable particles and make stable particles when together they form hadrons (e.g. a proton) which has charge $$ e $$. So in classical theory of electromagnetism, the smallest quantity of charge is $$ e $$.)

Electric and magnetic fields
Interaction among charges are described by electric and magnetic fields. Electric and magnetic fields are fully described by Maxwell's equations and suitable boundary conditions. The force on a charge in an electric field $$ \mathbf {E} $$ and magnetic field $$ \mathbf{B} $$ is given by


 * $$ \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf {B}) $$

This is known as Lorentz force law. From the mathematical description we see that the same fields have equal and opposite force on positive and negative charges of identical amount of charge and identical velocity.

The electric and magnetic fields can be derived from Maxwell's equations


 * $$\nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_0}$$


 * $$\nabla \cdot \mathbf{B}=0$$


 * $$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$


 * $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

and with boundary conditions that electric and magnetic field is zero at infinity. Lorentz force law and Maxwell's equations contains everything of the subject of electromagnetism.