Overview: Electromagnetic Waves

The waves in which electric and magnetic fields vary sinusoidally in mutually perpendicular directions and also perpendicular to the direction of propagation of the wave. Such waves which can actually propagate in space even without any material medium are called "electromagnetic waves".

The range of wavelengths of all these radiations is very large and on this basis they have been given an order. This order is called the 'electromagnetic spectrum'.

Average electric energy density:

\[\vec{u}=\frac{1}{4}\varepsilon_{0}E_{0}^{2}+\frac{1}{4}\frac{B_{0}^{2}}{\mu_{0}}\]

Average energy density in EM waves:

\[\vec{u}=\frac{1}{2}\varepsilon_{0}E_{0}^{2}=\frac{1}{2}\frac{B_{0}^{2}}{\mu_{0}}\cdot\]

\[<1>=\frac{1}{2}\varepsilon_{0}E_{0}^{2}c=\frac{1}{2}\frac{B_{0}E_{0}}{\mu_{0}}\]

Statement

Ampère’s circuital law states that the line integral of the magnetic field B around a closed loop is equal to μ₀ times the current enclosed by the loop. However, when the electric field changes with time, an additional current called the displacement current must be included. The modified law is known as the Ampère–Maxwell law.

Explanation

According to Ampère’s circuital law,

\[\oint\vec{B}\cdot d\vec{l}=\mu_0I\]

This law, in its present form, is not applicable to all situations. There are situations in which no conduction current exists, yet a magnetic field still exists.

Consider the charging of a capacitor connected to a battery. As current flows from the battery, the capacitor plates are charged, and an electric field develops between them. Because the current in the circuit varies with time, the electric field between the plates also varies.

At any instant, let the current in the circuit be I(t). Consider a point P₁​ at a distance r from the connecting wire. Applying Ampère’s law over a closed loop S₁​:

B(2πr) = μ₀I(t)   …(ii)

Now consider a point P₂​ between the capacitor plates and draw a similar loop S₂​. Since there is no conduction current between the plates:

B(2πr) = μ₀ × 0 = 0   …(iii)

This result implies that Ampère’s circuital law does not predict a magnetic field at P₂​, although a magnetic field is observed there. This inconsistency shows that the original form of Ampère’s law is incomplete.

The magnetic field at P₂​ is due to the change of electric field with time. If the plates of the capacitor have area A and charge q at any instant, the electric field between the plates is:

E = \[\frac {q}{ε_0A}\]​

The electric flux through surface S₂​ is:

Φ_E = E⋅A = \[\frac {q}{ε_0}\]   …(iv)

If the charge changes with time, there is a current I = \[\frac {dq}{dt}\]. Differentiating electric flux with respect to time:

\[\frac {dΦ_E}{dt}\] = \[\frac {1}{ε_0}\]\[\frac {dq}{dt}\]​

This term represents the current equivalent at a given point in the electric field and is called the displacement current.

Thus, the total current is the sum of conduction current I and displacement current I_d:

I = I + I_d = I + ε₀\[\frac {dΦ_E}{dt}\]   …(v)​

Conclusion

Both conduction current and displacement current produce magnetic fields. Therefore, the complete form of Ampère’s law, including displacement current, is called the Ampère–Maxwell law.

It states that the electric flux through any closed surface is equal to 1/ɛ₀ times the 'net' charge enclosed by the surface. Mathematically,

\[\oint\vec{\mathbf{E}}\cdot d\vec{\mathbf{S}}=\frac{q}{\varepsilon_0}\cdot\]

This equation is called Maxwell's first equation.

It states that the magnetic flux through any closed surface is zero. Mathematically,

∮\[\vec B\] ⋅ d\[\vec S\] = 0.

This equation is called Maxwell's second equation.

It states that an induced emf set up in a circuit is equal to the negative rate of change of magnetic flux through the circuit. Mathematically,

e = -\[\frac {dΦ_B}{dt}\].

Since, emf can be defined as the line integral of electric field,

\[\oint\vec{\mathbf{E}}\cdot d\vec{\mathbf{l}}=-\frac{d\Phi_{B}}{dt}\cdot\]

Thus, the law states that the line integral of electric field along a closed path is equal to the rate of change of magnetic flux through the surface bounded by that closed path. This is called Maxwell's third equation.

The ratio of the magnitudes of electric and magnetic fields equals the speed of light in free spасе.

v = \[\frac {1}{\sqrt{με}}\]

where u and & are the permeability and permittivity of the medium.

• Electromagnetic waves are produced by accelerated charges.
• They do not require a material medium and can propagate through a vacuum.
• In free space, they travel at the speed of light,
  c = 3.0 × 10⁸ m s⁻¹
• Electromagnetic waves are transverse; the electric and magnetic fields and the direction of propagation are mutually perpendicular.
• Electric and magnetic fields vary simultaneously and reach their maximum values, E0 and B0, at the same point in space and time.
• In free space, the magnitudes of the fields are related by
  \[\frac {E}{B}\] = c
• Energy is shared equally between electric and magnetic fields in an electromagnetic wave (on average).
• Electromagnetic waves carry energy and momentum and hence exert radiation pressure.
• Being uncharged, electromagnetic waves are not deflected by electric or magnetic fields.

• Newton discovered the visible spectrum of sunlight, extending from violet to red wavelengths.
• The Sun’s radiation extends beyond visible light, both below violet and above red, which are invisible to the human eye.
• Radiation below violet is called ultraviolet, while radiation above red is called infrared.
• Maxwell predicted that light is an electromagnetic wave, based on his four equations of electromagnetism.
• Different forms of electromagnetic radiation were discovered over time (radio waves, X-rays, gamma rays), all travelling in free space at the speed of light.