displacement current definition class 12 what is physics formula

Displacement Current

We know that a magnetic field changing with time gives rise to an emf (due to change in magnetic flux) and hence an electric field. Is the converse also true? James Clerk Maxwell (1831–1879) argued that this was indeed true, i.e. a time varying electric field gives rise to a magnetic field. From Ampere’s circuital law, this magnetic field will give rise to a current. Maxwell recognised that this current cannot be the conventional conduction current because it can exist even in a vacuum. He called this current the displacement current which exists in addition to the conduction current.

 Maxwell’s Equations

Maxwell modified Ampere’s circuital law by including the displacement current and formulated a set of equations involving electric and magnetic fields, their sources and charge and current densities. These equations are known as Maxwell’s equations, which are as follows :

where i_c = conduction current and i_d = displacement current. The other symbols have their usual meaning.

History of Electromagnetic Waves

Ampere’s law states that a time-varying electric field at any point is a source of magnetic field. Faraday’s law states just the reverse namely, that a time-varying magnetic field is a source of electric field. These laws led Maxwell to conclude that space and time varying electric and magnetic fields produce an electromagnetic disturbance which can travel even in particle-free space. This disturbance is called the electromagnetic wave. Thus, electromagnetic waves are those waves in which electric and magnetic fields vary sinusoidally in space and time. In 1865, Maxwell predicted the existence of electromagnetic waves. His theory further predicted that electromagnetic waves of all frequencies (and hence all wavelengths) should propagate with the speed of light. This theory was first experimentally verified by a German physicist, Heinrich Hertz in 1888. He used a simple electric oscillator and was able to pick up its radiation of electromagnetic waves on a radio receiver some distance away. In 1899, an Italian engineer Guglielmo Marconi succeeded in transmitting electromagnetic waves across the English Channel and 1901 across the Atlantic Ocean.

 Transverse Nature of Electromagnetic Waves

In an electromagnetic wave, the electric and magnetic fields are mutually perpendicular to each other and each field is perpendicular to the direction of propagation of the wave.

Velocity of Electromagnetic Waves

Maxwell’s theory predicted that electromagnetic waves of all frequencies (and hence all wavelengths) propagate in vacuum with a speed given by

where µ₀ is the magnetic permeability and ε₀ the electric permittivity of vacuum. Now, for vacuum, we know that µ₀ = 4π x 10^–7 TmA^–1 and ε₀ = 8.85 x 10^–12 C² N^–1m^–2, Substituting these values in Eq. (15.1) we have

which is the speed of light in vacuum measured experimentally. The excellent agreement between the experimentally measured speed of light (c = 2.997924 x 10⁸ ms^–1) to such a high degree of accuracy and the value based on the experimental measurements of ε₀ and µ₀ gave the first quantitative proof of the fact that light is an electromagnetic wave. The emergence of the speed of light from purely electromagnetic considerations is the crowning achievement of Maxwell’s electromagnetic theory. In a material medium, the speed of electromagnetic waves is given by

Production of Electromagnetic Waves

Consider a charge at rest. It has an electric field in the region around it but no magnetic field. If it is given an impulse so that it begins to move, it produces electric and magnetic fields. If it is moving with a constant velocity (i.e. if the current is not changing with time) the magnetic field will not change with time, so it cannot produce an electromagnetic wave. But if the charge is somehow accelerated, the magnetic and electric fields will change with space and time; it then produces an electromagnetic wave. Thus an accelerated charge emits an electromagnetic wave. In an L-C circuit, the charge oscillates between the capacitor plates. An oscillating charge has a non-zero acceleration; hence it will emit an electromagnetic wave whose frequency is the same that of the oscillating charge. An electron circulating round its nucleus in a stable orbit does not emit an electromagnetic wave, although it is accelerating; it does so only when it falls to a lower orbit. When fast-moving electrons hit a metal target, electromagnetic waves (X-rays) are produced.

Energy Density of Electromagnetic Field

Just as an oscillating pendulum has energy associated with it, oscillating electric and magnetic fields also have electric and magnetic energies associated with them. The average energy densities of electric and magnetic fields of an electromagnetic wave are respectively given by

i.e. in an electromagnetic wave, the average energy densities of electric and magnetic fields are equal. The total average energy density of the electromagnetic field is

Characteristics of Electromagnetic Waves

1. Electromagnetic waves are producted by an accelerating charge.

2. Electromagnetic waves can propagate even in vacuum.

3. They travel in vacuum with a speed of 3 x 10⁸ ms^–1, the speed of light.

4. Electromagnetic waves are transverse in nature, i.e. the oscillating electric and magnetic field vectors are perpendicular to the direction of propagation of the wave and are also perpendicular to each other

5. In an electromagnetic wave, the total energy of the electromagnetic field is shared equally between the electric and magnetic fields.

6. Electromagnetic waves of all frequencies exhibit the phenomena of interference, diffraction and polarization.

Spectrum of Electromagnetic Radiation Beginning with the remarkable demonstration by Hertz of the existence of long wavelength electromagnetic waves, scientists began looking for electromagnetic waves of wavelengths much shorter than the visible light. In 1898 Rontgen discovered X-rays which are electromagnetic waves of wavelength about 10^–10 m. The wavelength of visible light is in the range of 4 x 10^–7 to 8 x 10^–7 m. Radiowaves, X-rays and visible light are all electromagnetic waves and travel with the same speed c = 3.0 x 10⁸ ms^–1 in free space. They differ in wavelength (and hence in frequency, n = c/λ) only, which means that the sources that produce them and their detectors are different. Table 15.1 shows the frequency range, wavelength range, the names and the sources of the known electromagnetic radiations. The spectrum of electromagnetic radiation has no upper or lower limits and all the regions overlap. Notice that visible light is only a very small part of the total electromagnetic spectrum. We see that electromagnetic waves have a very wide of wavelengths (and hence of frequencies). Although they are identical in nature, their interaction with matter or their physiological action on living bodies depends on their frequency. Infrared rays are thermal radiations which produce heat, X-rays and gamma rays are highly penetrating, to mention only a few of the effects.

Uses of Electromagnetic Spectrum

The different regions of the total electromagnetic spectrum have been put to the following uses:

1. Radiowaves are used in radar and radio broadcasting.

2. Microwaves are used in long distance wireless communications via satellites.

3. Infrared, visible and ultraviolet traditions are used to know the structure of molecules.

4. Diffraction of X-rays by crystals gives the details of the structure of crystals.

5. The bones are opaque to X-rays but flesh is transparent. X-ray pictures of a human body are used in medical diagnosis of fractures and cracks of bones.

6. The γ -rays are used in the study of the structure of the nuclei of atoms.