**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

Source of electromagnetic waves

Maxwell’s theory of electromagnetic radiation

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

Nature of electromagnetic waves

Properties of electromagnetic waves

Energy in electromagnetic waves

**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,

$$\oint \stackrel{\u20d7}{\mathrm{B}}.\mathrm{d}\stackrel{\u20d7}{\mathrm{l}}={\mathrm{}\mathrm{\mu}}_{\mathrm{o}}\mathrm{i}\left(\mathrm{t}\right)$$

- 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) = μ

_{o}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****μ**_{0 }**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,

$${\mathrm{\u0424}}_{\mathrm{E}}=\left|\mathrm{E}\right|\mathrm{A}=\mathrm{}\frac{\mathrm{q}}{{\mathrm{\varepsilon}}_{\mathrm{o}}}$$

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

$${i}_{d}=\frac{dq}{dt}$$

Now,

$$\mathrm{}\frac{{\mathrm{d}\mathrm{\u0424}}_{\mathrm{E}}}{\mathrm{d}\mathrm{t}}=\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left(\frac{\mathrm{q}}{{\mathrm{\varepsilon}}_{\mathrm{o}}}\right)$$

$$=\frac{1}{{\mathrm{\varepsilon}}_{\mathrm{o}}}\mathrm{}\frac{\mathrm{d}\mathrm{q}}{\mathrm{d}\mathrm{t}}=\mathrm{}\frac{1}{{\mathrm{\varepsilon}}_{\mathrm{o}}}\mathrm{}{\mathrm{i}}_{\mathrm{d}}$$

$$\Rightarrow \mathrm{}\mathrm{}{\mathrm{i}}_{\mathrm{d}}=\mathrm{}{\mathrm{\varepsilon}}_{\mathrm{o}}\left(\frac{{\mathrm{d}\mathrm{\u0424}}_{\mathrm{E}}}{\mathrm{d}\mathrm{t}}\right)$$

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 *i*_{c}, and the displacement current denoted by,

$${\mathrm{i}}_{\mathrm{d}}={\mathrm{\varepsilon}}_{\mathrm{o}}\left(\frac{{\mathrm{d}\mathrm{\u0424}}_{\mathrm{E}}}{\mathrm{d}\mathrm{t}}\right)$$

That is,

$$\mathrm{i}={\mathrm{i}}_{\mathrm{c}}+\mathrm{}{\mathrm{i}}_{\mathrm{d}}={\mathrm{i}}_{\mathrm{c}}+\mathrm{}{\mathrm{\varepsilon}}_{\mathrm{o}}\left(\frac{{\mathrm{d}\mathrm{\u0424}}_{\mathrm{E}}}{\mathrm{d}\mathrm{t}}\right)\mathrm{}$$

The Amperes circuital law is modified as follows,

$$\Rightarrow \oint \mathrm{B}.\mathrm{d}\mathrm{l}={\mathrm{\mu}}_{\mathrm{o}}\mathrm{}\mathrm{i}\mathrm{}\left(\mathrm{t}\right)$$

$$=\mathrm{}{\mathrm{\mu}}_{\mathrm{o}}{\mathrm{i}}_{\mathrm{c}}+\mathrm{}{\mathrm{\mu}}_{\mathrm{o}}{\mathrm{\varepsilon}}_{\mathrm{o}}\left(\frac{{\mathrm{d}\mathrm{\u0424}}_{\mathrm{E}}}{\mathrm{d}\mathrm{t}}\right)$$

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 |
$$\oint \mathrm{E}.\mathrm{d}\mathrm{l}=\mathrm{}\left(\frac{{\mathrm{d}\mathrm{\u0424}}_{\mathrm{B}}}{\mathrm{d}\mathrm{t}}\right)\mathrm{}$$ |
Faraday’s law |

$$\oint \mathrm{B}.\mathrm{d}\mathrm{A}=0$$ |
Gauss’s Law for magnetism |
$$\oint \mathrm{B}.\mathrm{d}\mathrm{l}=\mathrm{}{\mathrm{\mu}}_{\mathrm{o}}{\mathrm{i}}_{\mathrm{c}}+\mathrm{}{\mathrm{\mu}}_{\mathrm{o}}{\mathrm{\varepsilon}}_{\mathrm{o}}\left(\frac{{\mathrm{d}\mathrm{\u0424}}_{\mathrm{E}}}{\mathrm{d}\mathrm{t}}\right)\mathrm{}$$ |
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 6×10^{14} Hz, while the maximum frequency of even with modern electronic circuits is about 10^{11} 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

$${\mathrm{E}}_{\mathrm{x}}=\mathrm{}{\mathrm{E}}_{\mathrm{o}}\mathrm{}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{}(\mathrm{k}\mathrm{z}\u2013\mathrm{\omega}\mathrm{t}\mathrm{})$$

$$\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}{\mathrm{B}}_{\mathrm{y}}=\mathrm{}{\mathrm{B}}_{\mathrm{o}}\mathrm{}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{}(\mathrm{k}\mathrm{z}\u2013\mathrm{\omega}\mathrm{t}),\mathrm{}$$

Here ω =2πν is the angular frequency and k = $\frac{2\pi}{\lambda}$ is the magnitude of the wave vector (or propagation vector) $\stackrel{\u20d7}{\mathrm{k}}$. Direction of $\stackrel{\u20d7}{\mathrm{k}}$ describes the direction of propagation of the wave.

We can see that speed of propagation of the wave is

$$\mathrm{c}=\mathrm{}\mathrm{\lambda}\mathrm{\nu}=\frac{\omega}{2\mathrm{\pi}}\times \frac{2\pi}{\lambda}=\left(\frac{\omega}{k}\right)$$

$$\Rightarrow \mathrm{\omega}\mathrm{}=\mathrm{}\mathrm{c}\mathrm{}\mathrm{k},\mathrm{}\mathrm{}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{}\mathrm{c}\mathrm{}=\frac{1}{\sqrt{{\mathrm{\mu}}_{\mathrm{o}}{\mathrm{\epsilon}}_{\mathrm{o}}}}\mathrm{}$$

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

$$\mathrm{c}=\mathrm{}\frac{{\mathrm{E}}_{\mathrm{o}}}{{\mathrm{B}}_{\mathrm{o}}}$$

Where E_{o} and B_{o} are maximum values of electric and magnetic field vectors.

**Properties of electromagnetic waves**

- Electromagnetic waves are produced by accelerated charged particles.
- These waves are transverse in nature.
- These waves propagate through space and do not need a medium
- The speed of electromagnetic wave in vacuum,
$$\mathrm{c}\mathrm{}=\frac{1}{\sqrt{{\mathrm{\mu}}_{\mathrm{o}}{\mathrm{\epsilon}}_{\mathrm{o}}}}\mathrm{}\mathrm{o}\mathrm{r}\mathrm{}\mathrm{c}\mathrm{}=\mathrm{}\frac{{\mathrm{E}}_{\mathrm{o}}}{{\mathrm{B}}_{\mathrm{o}}}$$

Where E

_{o}and B_{o}are maximum values of electric and magnetic field vectors.The velocity of light in a medium,

$$\mathbf{}\mathrm{v}\mathrm{}=\frac{1}{\sqrt{\mathrm{\mu}\mathrm{\epsilon}}}\mathrm{}$$

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 3×10

^{8}m/s

- 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,
$$\mathrm{}\mathrm{S}\mathrm{}=\frac{1}{{\mathrm{\mu}}_{\mathrm{o}}}\mathrm{E}\times \mathrm{B}$$

SI unit of S is watt/m

^{2}.Its magnitude S is defined as the rate at which energy is transported by a wave across a unit area at any instant.

- 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.
- The energy in electromagnetic waves is divided equally between electric field and magnetic field vectors.
The average electric energy density.

$${\mathrm{U}}_{\mathrm{E}}\mathrm{}=\mathrm{}\frac{{\mathrm{\epsilon}}_{\mathrm{o}}{\mathrm{E}}^{2}\mathrm{}}{2}$$

The average magnetic energy density,

$${\mathrm{U}}_{\mathrm{B}}\mathrm{}=\mathrm{}\frac{{\mathrm{B}}^{2}\mathrm{}}{2{\mathrm{\mu}}_{\mathrm{o}}}$$

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.

- 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,

$$\mathrm{p}\mathrm{}=\frac{\mathrm{U}}{\mathrm{c}}$$

- The energy in electromagnetic waves is divided equally between electric field and magnetic field vectors.

**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.

**Radio waves**

Radio waves are produced by the accelerated motion of charges in conducting wires. They are used in radio and television communication systems. They are frequency range from 500 kHz to 1000 MHz.

The AM (amplitude modulated) band is from 530 kHz to 1710 kHz. Higher frequencies upto 54 MHz are used for **short wave** bands. TV waves range from 54 MHz to 890 MHz. The FM (frequency modulated) radio band extends from 88 MHz to 108 MHz. Cellular phones use radio waves to transmit voice communication in the ultrahigh frequency (UHF) band.

**Microwaves**

Microwaves (short-wavelength radio waves), with frequencies in the gigahertz (GHz) range are produced by special vacuum tubes (called klystrons, magnetrons and Gunn diodes). Due to their short wavelengths, they are suitable for the radar systems used in aircraft navigation.

Radar also provides the basis for the speed guns used to time fast balls, tennis serves, and automobiles.

In microwave ovens the frequency of the microwaves is selected to match the resonant frequency of water molecules so that energy from the waves is transferred efficiently to the kinetic energy of the molecules. This raises the temperature of any food containing water.