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

Maxwell’s generalisation of Ampere’s circuital law

Maxwell equations

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

Speed of electromagnetic wave

Poynting vector

Energy in electromagnetic waves

Radiation pressure

Electromagnetic spectrum

Radio 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,

B.dl= μoi(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,

ФE=EA= qɛo

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

id= dqdt



=1ɛo dqdt= 1ɛo id

  id= ɛodФEdt

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,


That is,

i=ic+ id=ic+ ɛodФEdt 

The Amperes circuital law is modified as follows,

B.dl=μo i t

= μoic+ μoɛodФEdt

This is called Ampere Maxwell law.

Maxwell equations


Gauss’s Law for electricity

E.dl= dФBdt 

Faraday’s law


Gauss’s Law for magnetism

B.dl= μoic+ μoɛodФEdt 

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

Ex= Eo sin (kzωt )

and By= Bo sin (kzωt), 

Here ω =2πν is the angular frequency and k = 2πλ is the magnitude of the wave vector (or propagation vector) k. Direction of k describes the direction of propagation of the wave.

We can see that speed of propagation of the wave is

c= λν=ω2π×2πλ=ωk

ω = c k,  where c =1μoεo 

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

c= EoBo

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,

    c =1μoεo or c = EoBo

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

    The velocity of light in a medium,

    v =1με 

    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,

     S =1μoE×B

    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.

      UE = εoE2 2

      The average magnetic energy density,

      UB = B2 2μo

    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,

    p =Uc

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 (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.