Electromagnetic Waves: Comprehensive NEET Physics Notes

1. Introduction to Electromagnetic Waves

1.1 Key Concepts and Formulae

Maxwell's Equations: These equations form the foundation of electromagnetism and predict the existence of electromagnetic waves.
1. Gauss's Law for Electricity: $\nabla \cdot E = \frac{ρ}{ϵ _{0}}$
2. Gauss's Law for Magnetism: $\nabla \cdot B = 0$
3. Faraday's Law of Induction: $\nabla \times E = - \frac{\partial B}{\partial t}$
4. Ampere-Maxwell Law: $\nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \frac{\partial E}{\partial t}$

1.2 Displacement Current

Formula: Displacement current is given by: $I_{d} = ϵ_{0} \frac{d Φ _{E}}{d t}$
- Explanation: This concept is introduced to account for the changing electric field between capacitor plates, ensuring the consistency of Ampere's Law in time-varying fields.

2. Electromagnetic Wave Propagation

2.1 Nature of Electromagnetic Waves

Formulae:
Explanation: The electric and magnetic fields in electromagnetic waves are perpendicular to each other and to the direction of propagation. They oscillate in phase, meaning the peaks and troughs of these fields occur simultaneously.
- Electric field: $E = E_{0} sin (k z - ω t) \overset{x}{^}$
- Magnetic field: $B = B_{0} sin (k z - ω t) \overset{y}{^}$
- Speed of light: $c = \frac{1}{μ _{0} ϵ _{0}}$

2.2 Electromagnetic Spectrum

Wavelength Range: Electromagnetic waves have a broad spectrum, categorized as follows:
Type of Wave
Wavelength Range
Radio Waves
$λ > 0.1 m$
Microwaves
$1 mm < λ < 0.1 m$
Infrared
$700 nm < λ < 1 mm$
Visible Light
$400 nm < λ < 700 nm$
Ultraviolet
$10 nm < λ < 400 nm$
X-rays
$0.01 nm < λ < 10 nm$
Gamma Rays
$λ < 0.01 nm$

Type of Wave	Wavelength Range
Radio Waves	$λ > 0.1 m$
Microwaves	$1 mm < λ < 0.1 m$
Infrared	$700 nm < λ < 1 mm$
Visible Light	$400 nm < λ < 700 nm$
Ultraviolet	$10 nm < λ < 400 nm$
X-rays	$0.01 nm < λ < 10 nm$
Gamma Rays	$λ < 0.01 nm$

3. Example Applications

3.1 Example Problem 1: Calculating Magnetic Field from Electric Field

Problem: A plane electromagnetic wave with a frequency of 25 MHz travels in free space along the x-direction. If at a particular point in space and time, the electric field is given as $E = 6.3 \hat{j} V / m$ , calculate the magnetic field at this point.
Solution:
- Step 1: Use the relation $B = \frac{E}{c}$ .
- Step 2: Substitute the given values: $B = \frac{6.3 V / m}{3 \times 1 0 ^{8} m / s} = 2.1 \times 1 0^{- 8} T$ .
- Step 3: Determine the direction of the magnetic field, which should be perpendicular to both the electric field and the direction of wave propagation.

3.2 Example Problem 2: Determining Wavelength and Frequency

Problem: The magnetic field in a plane electromagnetic wave is given by $B_{y} = 2 \times 1 0^{- 7} sin (0.5 \times 1 0^{3} x + 1.5 \times 1 0^{11} t)$ . Find the wavelength and frequency of the wave.
Solution:
- Step 1: Identify the wave number $k = 0.5 \times 1 0^{3} m^{- 1}$ .
- Step 2: Use the relation $λ = \frac{2 π}{k}$ to find the wavelength.
- Step 3: Identify the angular frequency $ω = 1.5 \times 1 0^{11} r a d / s$ .
- Step 4: Use $f = \frac{ω}{2 π}$ to find the frequency.

4. Common Mistakes

4.1 Misunderstanding the Displacement Current

Mistake: Students often confuse displacement current with conduction current, not realizing it arises in a region where there is a time-varying electric field but no actual movement of charges.
Tip: Always remember that displacement current is a concept introduced to maintain consistency in Maxwell's equations and does not represent the flow of real charges.

4.2 Incorrect Use of Speed of Light Formula

Mistake: Forgetting that the speed of light in a medium is given by $v = \frac{1}{μ ϵ}$ and not by the free space value $c$ .
Tip: Ensure that you substitute the correct values for the permittivity $ϵ$ and permeability $μ$ of the medium when calculating the speed of light in a medium.

Quick Recap

Maxwell's Equations unify electricity and magnetism and predict electromagnetic waves.
Displacement Current is a key concept introduced by Maxwell to address inconsistencies in Ampere's law.
Electromagnetic Waves are transverse waves where electric and magnetic fields oscillate perpendicularly.
Electromagnetic Spectrum spans from radio waves to gamma rays, classified by wavelength.

Practice Questions

A plane electromagnetic wave propagates in vacuum along the z-axis. If the electric field is given by $E_{x} = 5 sin (2 π \times 1 0^{6} t - k z) V / m$ , find the corresponding magnetic field.
Calculate the displacement current in a parallel plate capacitor with a capacitance of 5 μF, where the voltage across the plates changes according to $V (t) = 10 sin (100 π t) V$ .

These questions will reinforce your understanding of how electromagnetic waves propagate and how displacement current is calculated in time-varying electric fields.