CBSE NOTES CLASS 12 PHYSICS

CHAPTER 8 ELECTROMAGNETIC WAVES

Displacement current - the missing term in Ampere’s circuital law

Maxwell’s generalisation of Ampere’s circuital law

Maxwell equations

Source of electromagnetic waves

Why can’t we prove that light is an electromagnetic wave?

Nature of electromagnetic waves

Properties of electromagnetic waves

Speed of electromagnetic wave

Poynting vector

Electromagnetic spectrum

Microwaves

CBSE NOTES CLASS 12 PHYSICS

CHAPTER 8 ELECTROMAGNETIC WAVES

Displacement current – the missing term in Ampere’s circuital law

An electrical current produces a magnetic field around it. Maxwell showed that for logical consistency, a changing electric field must also produce a magnetic field.

Let us consider the process of charging of a capacitor. According to Ampere’s circuital law,

• Let us first find the magnetic field at a point P, outside the parallel plate capacitor, s shown in the diagram.

For this, we consider a plane circular loop of radius r whose plane is perpendicular to the direction of the current-carrying wire, and which is centred symmetrically with respect to the wire. We can see that the magnetic field is directed along the circumference of the circular loop and is the same in magnitude at all points on the loop. If B is the magnitude of the field, the left side of equation becomes B (2πr), i.e.,

B (2πr) = μo i(t)

• Now let us consider a different surface, which has the same boundary, but different shape This is a pot like surface or shaped like a tiffin box (without the lid). which nowhere touches the current, but has its bottom between the capacitor plates; its mouth is the circular loop.

On applying Ampere’s circuital law to such surfaces with the same perimeter, we find that the left hand side of the equation has not changed but the right hand side is zero (no current passes through the surface) and not μ0 i (t).

This leads to contradiction giving different magnetic field at the same point.

Maxwell pointed out that this is due to the missing term corresponding to electrical field.

Now considering the surface in second case, the electrical field is perpendicular to the surface ‘S’. It has the same magnitude over the area ‘A’ of the capacitor plates, and is zero outside it.

Using Gauss’s law,

If the charge q on the capacitor plates changes with time, there is a current,

Now,

Maxwell pointed out that this is the missing term in Ampere’s circuital law

Maxwell’s generalisation of Ampere’s circuital law

The source of a magnetic field is not just the conduction electric current, but also the time rate of change of electric field. Or the total current i is the sum of the conduction current denoted by ic, and the displacement current denoted by,

${\mathrm{i}}_{\mathrm{d}}={\mathrm{\varepsilon }}_{\mathrm{o}}\left(\frac{{\mathrm{d}\mathrm{Ф}}_{\mathrm{E}}}{\mathrm{d}\mathrm{t}}\right)$

That is,

The Amperes circuital law is modified as follows,

This is called Ampere Maxwell law.

Maxwell equations

 $\oint \mathrm{E}.\mathrm{d}\mathrm{A}=\frac{\mathrm{q}}{{\mathrm{\varepsilon }}_{\mathrm{o}}}$ Gauss’s Law for electricity Faraday’s law $\oint \mathrm{B}.\mathrm{d}\mathrm{A}=0$ Gauss’s Law for magnetism Ampere-Maxwell law

Source of electromagnetic waves - Maxwell’s theory of electromagnetic radiation

A stationary charge produces only electrostatic fields, while the charges in uniform motion (steady currents) produce magnetic fields that do not vary with time.

According to Maxwell’s theory accelerated charges radiate electromagnetic waves.

Let us consider a charge oscillating with some frequency (it is an accelerating charge). This produces an oscillating electric field in space, which produces an oscillating magnetic field, which in turn, is a source of oscillating electric field, and so on. The oscillating electric and magnetic fields thus regenerate each other, as the wave propagates through the space.

The frequency of the electromagnetic wave equals the frequency of oscillation of the charge. The energy associated with the propagating wave comes at the expense of the energy of the source – the accelerated charge.

Why can’t we prove that light is an electromagnetic wave?

It is not possible to set up an ac circuit in which the current oscillate at the frequency of visible light. The frequency of yellow light is about 6×1014 Hz, while the maximum frequency of even with modern electronic circuits is about 1011 Hz. This is why the experimental demonstration of electromagnetic wave could not be proved for light. Hertz in his experiment (1887) proved this for in the low frequency region of electromagnetic waves (the radio wave region).

Nature of electromagnetic waves

Electromagnetic waves are combination of oscillating magnetic and electrical fields which are mutually perpendicular to each other and also perpendicular to the propagation of the wave. Electric and magnetic field vectors change sinusoidally.

If the propagation of wave is taken to be in z direction, Electrical field along x direction and Magnetic field along y directions, then the equation of plane progressive electromagnetic wave can be written as

Here ω =2πν is the angular frequency and k = $\frac{2\pi }{\lambda }$ is the magnitude of the wave vector (or propagation vector) $\stackrel{⃗}{\mathrm{k}}$. Direction of $\stackrel{⃗}{\mathrm{k}}$ describes the direction of propagation of the wave.

We can see that speed of propagation of the wave is

The magnitude of the electric and the magnetic fields in an electromagnetic wave are related as

Where Eo and Bo are maximum values of electric and magnetic field vectors.

Properties of electromagnetic waves

1. Electromagnetic waves are produced by accelerated charged particles.

2. These waves are transverse in nature.

3. These waves propagate through space and do not need a medium

4. The speed of electromagnetic wave in vacuum,

Where Eo and Bo are maximum values of electric and magnetic field vectors.

The velocity of light in a medium,

where, μ= relative permeability and ε = electrical permittivity of the medium.

Thus, the velocity of light depends on electric and magnetic properties of the medium.

The velocity of electromagnetic waves in free space or vacuum is an important fundamental constant and is same for all electromagnetic waves. Speed of light is vacuum is 3×108 m/s

1. The rate of flow of energy in an electromagnetic wave is described by the vector S called the poynting vector, which is, defined by the expression,

SI unit of S is watt/m2.

Its magnitude S is defined as the rate at which energy is transported by a wave across a unit area at any instant.

2. The electromagnetic waves carry energy from one place to another. The radio and TV signals from broadcasting stations carry energy. Light carries energy from the sun to the earth, thus making life possible on the earth.
1. The energy in electromagnetic waves is divided equally between electric field and magnetic field vectors.

The average electric energy density.

The average magnetic energy density,

The electric vector is responsible for the optical effects of an electromagnetic wave. Intensity of electromagnetic wave is defined as energy crossing per unit area per unit time perpendicular to the directions of propagation of electromagnetic wave.

1. An electromagnetic wave (like other waves) carries both energy and momentum. Since it carries momentum, an electromagnetic wave also exerts pressure, called radiation pressure.

If the total energy transferred to a surface in time t is U, the magnitude of the total momentum delivered to this surface (for complete absorption) is given by,

Electromagnetic spectrum

Electromagnetic waves include visible light waves, X-rays, gamma rays, radio waves, microwaves, ultraviolet and infrared waves. The classification of electromagnetic waves according to frequencies is referred to as the electromagnetic spectrum. There is no sharp division between one kind of wave and the next. The classification is based roughly on how the waves are produced and/or detected.