Electromagnetic Waves
Electromagnetic waves are produced due to time-varying electric and magnetic fields and propagate through space without the need for a material medium. The theoretical foundation of electromagnetic waves was established by Maxwell through a set of four fundamental equations, known as Maxwell's equations. These equations explain how electric and magnetic fields are generated by charges, currents, and changing fields, and how they sustain each other during wave propagation. Understanding Maxwell's equations and the concept of displacement current is essential for explaining the origin, propagation, and properties of electromagnetic waves.
Maxwell's Equations
James Clerk Maxwell formulated these relations in the 19th century. Together they explain static and time-varying electric and magnetic fields, and predict electromagnetic waves (light). Each equation has an integral form (useful for symmetry arguments and boundary conditions) and a differential form (local form used in field theory).
James Maxwell1. Gauss's Law for Electricity
Integral form: ∮ E · dA = Q_enclosed / ε0
Differential form: ∇ · E = ρ / ε0
- E is the electric field (vector).
- ρ is the volume charge density.
- ε0 is the permittivity of free space.
- The law states that the net electric flux through a closed surface equals the total enclosed charge divided by ε0.
2. Gauss's Law for Magnetism
Integral form: ∮ B · dA = 0
Differential form: ∇ · B = 0
- B is the magnetic field (vector).
- This law means magnetic field lines are continuous closed loops and there are no isolated magnetic charges (no magnetic monopoles observed).
3. Faraday's Law of Electromagnetic Induction
Integral form: ∮ E · dl = - d/dt ∫ B · dA
Differential form: ∇ × E = - ∂B / ∂t
- A time-varying magnetic field produces a circulating electric field.
- This is the principle behind transformers, electric generators and induced emf in coils.
4. Ampère's Law with Maxwell's Addition (Ampère-Maxwell Law)
Ampère's law relates the magnetic field around a closed loop to the electric current passing through the loop. Maxwell added a term to this law to account for the displacement current, making it consistent with the continuity equation. The modified equation is:
where J is the current density and
Try yourself: Which set of equations comprehensively describes the behavior of electric and magnetic fields?
Displacement Current
Why Maxwell introduced displacement current: Consider the continuity equation ∇ · J + ∂ρ/∂t = 0 which expresses charge conservation. If Ampère's law had only the conduction current term μ0J, taking divergence of both sides would give a contradiction unless ∂ρ/∂t = 0. To remove this inconsistency Maxwell added the term μ0ε0 ∂E/∂t so that the modified law is compatible with charge conservation in regions where charges accumulate (for example, between capacitor plates).
- Definition: The displacement current density is JD = ∂D / ∂t, where D is the electric displacement field. In vacuum D = ε0 E and, in a linear dielectric, D = ε E.
- Displacement current: The quantity that contributes to the magnetic field in regions where there is a time-varying electric field but no actual charge carriers moving. Its unit is ampere per square metre (A m-2).
- Between capacitor plates: When a capacitor charges, conduction current flows in the wires but not across the insulating gap. The changing electric field between the plates produces a displacement current which produces the same magnetic field as the conduction current in the wires, ensuring the continuity of magnetic field lines and consistency of Ampère's law.
Displacement current in a capacitorRelation between displacement current and total current across a surface:
ID = ∫ JD · dA = ∫ ∂D/∂t · dA = d/dt ∫ D · dA
- For a capacitor with plate area S and uniform field, ID = JD S = S (∂D / ∂t).
- Where S is the area of the capacitor plate, ID is the displacement current, JD is the displacement current density, D is related to E by D = ε E, and ε is the permittivity of the medium between the plates.
- Thus, displacement current is not the motion of charges across the dielectric; it represents the time rate of change of electric displacement and so completes the symmetry with magnetic effects.
Applications
- Charging capacitor: In the circuit branch containing a charging capacitor, conduction current exists in the external wires, while between the capacitor plates the changing electric field corresponds to a displacement current. Both give the same magnetic effect in surrounding space.
- Electromagnetic waves: Maxwell's addition made the set of equations symmetric and led to wave solutions in free space; the speed of these waves is c = 1 / √(μ0 ε0), identified with the speed of light.
- Practical devices: Faraday's law underlies transformers and generators; the Ampère-Maxwell law is essential in the design of antennas and understanding radiation from time-varying currents.
Electromagnetic Waves
Electromagnetic waves are those waves in which electric and magnetic field vectors changes sinusoidally and are perpendicular to each other as well as at right angles to the direction of propagation of wave.
The equation of plane progressive electromagnetic wave can be written as
E = Eo sin Ω (t - x / c) and B = Bo sin Ω (t - x / c). Where, Ω =2πv
Electromagnetic waves are produced by accelerated charge particles.
Properties of EM Waves
(i) These waves are transverse in nature.
(ii) These waves propagate through space with speed of light, i.e., 3 * 108 m / s.
(iii) The speed of electromagnetic wave,
c = 1 / √μo εo
where, μo is permittivity of free space,
∴ c = Eo / Bo
where Eo and Bo are maximum values of electric and magnetic field vectors.
According to Maxwell, when a charged particle is accelerated, it produces electromagnetic wave. The total radiant flux at any instant is given by,
p = q2a2 / 6 πεoc2
(iv) 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 = 1 / μo E * B
SI unit of S is watt/m2.
(v) Its magnitude S is related to the rate at which energy is transported by a wave across a unit area at any instant.
(vi) The energy in electromagnetic waves is divided equally between electric field and magnetic field vectors.
(vii) The average electric energy density.
UE = 1 / 2 εo E2 = 1 / 4 εo E2o
(viii) The average magnetic energy density,
UB = 1 / 2 B2 / μo = 1 / B2o / μo
(ix) The electric vector is responsible for the optical effects of an electromagnetic wave.
(x) Intensity of electromagnetic wave is defined as energy crossing per unit area per unit time perpendicular to the directions of propagation of electromagnetic wave.
(xi) The intensity I is given by the relation,
I = < μ > c = 1 / 2 εo E2oc
(xii) The existence of electromagnetic waves was confirmed by Hertz experimentally in 1888.
Propagation of Electromagnetic Waves
In radio wave communication between two places. the electromagnetic waves are radiated out by the transmitter antenna at one place which travel through the space and reach the receiving antenna at the other place.
Electromagnetic Spectrum
The arranged array of electromagnetic radiations in the sequence of their wavelength or frequency is called electromagnetic spectrum
Radio and microwaves are used in radio and TV communication,
Infrared rays are used to
(i) Treat muscular straw.
(ii) For taking photographs' in fog or smoke.
(iii) In green house to keep plants warm.
(iv) In weather forecasting through infrared photography.
Ultraviolet rays are used
(i) In the study of molecular structure.
(ii) In sterilizing the surgical instruments.
(iii) In the detection of forged documents, Eringer prints.
X-rays are used
(i) In detecting faults, cracks, flaws and holes in metal products.
(ii) In the study of crystal structure.
(iii) For the detection of pearls in oysters.
γ - rays are used for the study of nuclear structure
Solved Examples on EM Waves
Question 1: Calculate the frequency and wavelength of an electromagnetic wave with an energy of 6.626×10-19 J.
Answer:
- Frequency (f):E/h=6.626×10-19/6.626×10-34=1015 Hz.
- Wavelength (λ): c/f=3×108/1015=3×10-7 meters.
Question 2: What are the applications of X-rays?
Answer: X-rays are used in medical diagnostics to detect bone fractures and other ailments. They are also useful for ionization purposes.
Question 3: Are X-rays and gamma rays suitable for broadcasting radio, TV, or mobile signals?
Answer: No, X-rays and gamma rays are unsuitable for broadcasting because they have short ranges and are harmful. Their high penetrating power can damage living tissue.
Question 4: Identify which of the following is not a property of electromagnetic waves:
- Momentum
- Energy
- Pressure
- Heat energy
Answer:
- 1. Electromagnetic (EM) waves can transfer momentum to materials they interact with.
- 2. EM waves carry energy and can transmit it through a vacuum.
- 3. EM waves exert pressure, as demonstrated by a radiometer, which spins due to light-induced pressure differences.
- 4. EM waves do not carry heat energy, but they can heat objects upon absorption.
Question 5: A sunlight ray passing through your kitchen window hits a prism, creating a rainbow on the windowsill. If you place a radiometer on a specific color of the rainbow and measure the energy as 4.0×10-19 joules, what color might you have measured? Use Planck's constant 6.6256×10-34 J·s to determine this.
Answer:
Equation to use: E=h⋅f
Solving for frequency (f): f=E/h
Given:
- Energy (E) = 4.0×10-19 joules
- Planck's constant (h) = 6.6256×10-34 J·s
Calculation: f=4.0×10-19 / 6.6256×10-34≈6.03×1014 Hz
This frequency falls within the visible spectrum, near the green color range. Therefore, the measured color is likely green, though it could also be cyan or blue.
FAQs on Electromagnetic Waves
| 1. What are Maxwell's Equations? |  |
| 2. What is the significance of Maxwell's Equations in physics? | |
| 3. What is the Displacement Current in Maxwell's Equations? | |
| 4. How does the Displacement Current relate to electromagnetic waves? | |
| 5. How do Maxwell's Equations and the Displacement Current impact technology today? | |
