Electromagnetic Waves
Incomplete Article / Work in Progress
Table of Contents
1. Introduction
From the study of interference, diffraction, and polarization phenomenon we know that light is a transverse wave that travels at the speed \(c=3*10^2 m/s^2\) in vaccum. Using the rules/laws of electromagnetism Maxwell found out that there can exist EM waves travelling with the velocity \(v = \frac{1} {\sqrt{\mu_0 \epsilon_0}}\) . This value coincided almost perfectly with hte measured value of c. This strongly suggested that light is a EM wave.
EM waves (Electro Magnetic Waves) are wave in the EM field. The EM field is a mathematical concept. In 2D it can be visualized as the surface of a big pond. When a stone is dropped there are oscillations on the surface and waves propagate. Similarly light is aslo oscillation in electric and magnetic fields. It is fluctuation of the EM field intensity. To each point in space a value of this intensity is assigned.
Let's take the concept of field in a simple example: Let's take a 1D string (suppose a guitar string). The length of string is the x-axis. sth At some instant, the vibrating string may be in this position (fig 1). This can be represented by assigning an amplitude to each point on the x-axis (string) : \(A(x)\) similar to how vibrations alter the string, an charge in space alters/creates the electric field around it. sth And instead of single scalar amplitude, the electric field intensity is a vector and is represented as : \(\vec{E}(x,y,z)\)
2. Operators
Operations regarding vector fields : divergence and curl, and scalar fields : gradient are taught in mathematics course in detail. So Here's a brief intuition about what each operation does.
2.1. Gradient
It is a operator that operates on a scalar field (\(\phi\)) and gives a vector field which shows where the scalar field is increasing the most at each point in space. For example, lets consider a potential function V of a point charge. We know that the electric field of that charge is \[\vec{E} = - \nabla V\] This means electric field points in the direction where the potential is decreasing the most (decreasing because of -ve sign). This is indeed what we had studied in pervious chapters. So here the Gradient operator computed a vector field (Electric field) from given scalar field (Potential)
2.2. Divergence \(\vec{\nabla}\cdot\vec{v}\)
It takes a vector field and returns a scalar field that gives a measure of the flow of the field or the increase or decrease in the flux of the field. For example, lets take a charge distribution as shown below: sth If we consider region 1: net electric flux is +ve, it is +ve if electric field is flowing out or orginating from the region. So, the total/net divergence in region 1 is +ve, in region 2 is 0 and box 3 is -ve. From Gauss's Law, we know:
\[\iint{\vec{E} \cdot \mathop{d\vec{A}}} = \frac{q} {\epsilon_0}\]
Also, using Gauss's Divergence theorem, above equation can be written as:
\[\iiint{(\vec{\nabla}\cdot\vec{E}) \mathop{dV}} = \frac{q} {\epsilon_0}\]
The first equation says that the total field lines (flux) coming out of the surface A is \(\frac {q} \epsilon_0\). And the second tells that this value is also equal to the total electric field originating (\(\vec{\nabla}\cdot\overrightarrow{E}\)) inside the volume (V) enclosed by the surface A. This might make an intuitive sense to you that the net amount of field that orignates in a volume equals to the total amount of flux that comes out of the surface enclosing the volume. This is how we should interpretate Gauss's Divergence theorem.
2.3. Curl
3. Maxwell's Equations
3.1. Gauss's Law for Electric Field
\[\oint{\vec{E} \cdot \mathop{d\overrightarrow{S}}} = \frac {q_{enc}} {\epsilon_0}\] where \(q_{enc} =\) net charge enclosed.
