Chapter 8: Electromagnetic Waves
1. Displacement Current
Introduction: The concept of displacement current solves the inconsistency in Ampere's Circuital Law when applied to a capacitor being charged.
Derivation: Concept of Displacement Current
1. Logical Inconsistency: Consider a parallel plate capacitor charging.
Apply Ampere's Law $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$ to two loops:
- Loop 1 (Outside plates): Encloses conduction current ($I_c$). Magnetic field exists.
- Loop 2 (Between plates): Encloses NO conduction current ($I=0$). Predicted magnetic field is zero.
2. The Contradiction: Experimentally, a magnetic field does exist between the plates. Maxwell concluded that a changing electric field acts as a source of magnetic field.
3. The Formula: The current due to changing electric flux ($\phi_E$) is:
• $I_d$: Displacement Current (Ampere)
• $\epsilon_0$: Permittivity of free space ($8.85 \times 10^{-12} \, C^2/Nm^2$)
• $\frac{d\phi_E}{dt}$: Rate of change of electric flux (Vm/s)
4. Physical Significance: Displacement current indicates that time-varying electric fields produce magnetic fields. It is not a flow of charge but has the same magnetic effect as conduction current.
2. Maxwell's Equations
Maxwell unified electricity and magnetism into four fundamental equations.
| Name | Equation | Physical Significance |
|---|---|---|
| 1. Gauss’s Law (Electrostatics) | $\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q}{\epsilon_0}$ | Electric charge is the source of electric field. Field lines start/end on charges. |
| 2. Gauss’s Law (Magnetism) | $\oint \mathbf{B} \cdot d\mathbf{A} = 0$ | Isolated magnetic monopoles do not exist. Magnetic field lines form closed loops. |
| 3. Faraday’s Law | $\oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\phi_B}{dt}$ | Changing magnetic field induces an electric field (EMF). |
| 4. Ampere-Maxwell Law | $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 (I_c + \epsilon_0 \frac{d\phi_E}{dt})$ | Magnetic field is produced by conduction current AND changing electric field. |
3. Electromagnetic Waves
• Charges at rest $\rightarrow$ Electric field only.
• Charges in uniform motion $\rightarrow$ Magnetic field.
• Accelerated Charges $\rightarrow$ EM Waves (Electric + Magnetic Fields radiating energy).
Definition: EM waves are transverse waves consisting of oscillating electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields perpendicular to each other and to the direction of propagation.
• Valid strictly in Vacuum.
• In medium, replace $\mu_0, \epsilon_0$ with $\mu, \epsilon$.
• $E_0, B_0$ are peak values (Amplitudes).
4. Electromagnetic Spectrum
[attachment_0](attachment)| Type | Wavelength ($\lambda$) | Production Source | Key Applications |
|---|---|---|---|
| Radio | $> 0.1$ m | Accelerating charges in wires | Communication (TV, Radio) |
| Microwave | $0.1$ m - $1$ mm | Klystron/Magnetron valves | Radar, Ovens, Speed Guns |
| Infrared | $1$ mm - $700$ nm | Vibration of atoms (Heat) | Remote, Night Vision, Haze Photo |
| Visible | $700$ nm - $400$ nm | Atomic electron transitions | Vision, Optical Instruments |
| Ultraviolet | $400$ nm - $1$ nm | Inner shell electrons, Sun | Water Purifier (RO), LASIK |
| X-Rays | $1$ nm - $10^{-3}$ nm | Electron bombardment | Fracture detection, Crystal study |
| Gamma | $< 10^{-3}$ nm | Radioactive decay | Cancer Treatment (Radiotherapy) |
5. Question Bank (Mission 100)
A. Multiple Choice Questions (Sample)
-
Q. Which of the following has the minimum wavelength?
A) X-rays B) Gamma rays C) Microwaves D) Radio waves
Ans: B) Gamma rays -
Q. The quantity $\sqrt{\mu_0 \epsilon_0}$ represents:
A) Speed of light B) Inverse of speed of light C) Impedance D) Energy
Ans: B) Inverse of speed of light ($1/c$) -
Q. Displacement current is caused by:
A) Flow of electrons B) Changing magnetic field C) Changing electric field D) Steady electric field
Ans: C) Changing electric field -
Q. Waves used in Radar systems are:
A) Infrared B) Ultraviolet C) Microwaves D) X-Rays
Ans: C) Microwaves -
Q. In an EM wave, the phase difference between E and B fields is:
A) 0 B) $\pi/2$ C) $\pi$ D) $\pi/4$
Ans: A) 0 (They are in same phase)
B. Very Short Answer (VSA)
- Q. Write the expression for the speed of EM waves in a vacuum.
Ans: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ - Q. Which part of the spectrum is used in 'Greenhouse Effect'?
Ans: Infrared rays. - Q. Name the physical quantity which remains conserved for EM waves.
Ans: Momentum and Energy. - Q. What is the ratio of $E_0$ to $B_0$ in an EM wave?
Ans: $E_0/B_0 = c$ (Speed of light). - Q. Who experimentally verified the existence of EM waves?
Ans: Heinrich Hertz (1887).
C. Short Answer Type
Q1. Why are microwaves used in Radar systems?
Ans: Microwaves have short wavelengths (high frequency), which allows them to travel in straight lines without much diffraction. They can penetrate atmosphere and clouds effectively, making them ideal for radar and communication.
Q2. State the condition under which a displacement current is produced.
Ans: Displacement current ($I_d$) is produced whenever the electric flux through a given surface changes with time. $I_d = \epsilon_0 (d\phi_E / dt)$.
Q3. An EM wave travels in the z-direction. If the electric field is in the x-direction, what is the direction of the magnetic field?
Ans: The direction of propagation is $\mathbf{E} \times \mathbf{B}$. Since wave is in $\mathbf{k}$ (z-axis) and $\mathbf{E}$ is in $\mathbf{i}$ (x-axis), then $\mathbf{B}$ must be in $\mathbf{j}$ (y-axis) because $\mathbf{i} \times \mathbf{j} = \mathbf{k}$.
D. Long Answer Type (Derivation Focus)
Q1. Establish the inconsistency in Ampere's circuital law and hence deduce the expression for displacement current.
(Student Tip: Follow the derivation SOP provided in Section 1 of these notes. Draw the capacitor loop diagram clearly.)
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