MP Board · Class 12 · Physics · Electromagnetic WavesWhat are electromagnetic waves? Discuss their characteristics and write down the electromagnetic spectrum in order of increasing frequency, mentioning one use of each type. Also, solve the following numerical problem: A plane electromagnetic wave travels in vacuum along the z-direction. Suppose the electric field vector is $Ex = 3.1 \times 10^4 \text{ V/m}$ and the magnetic field vector $By$ is at a given point. Calculate the magnitude of magnetic field vector $By$ and the total energy density of the electromagnetic wave.

Definition of Electromagnetic Waves\nElectromagnetic waves are those waves in which electric and magnetic fields are oscillating perpendicularly to each other as well as to the direction of propagation of the wave. They are produced by accelerated charges and do not require any material medium for their propagation.

Characteristics of Electromagnetic Waves

• Transverse Nature: Electromagnetic waves are transverse in nature because the electric and magnetic field vectors vibrate perpendicular to the direction of wave propagation.
• Speed: In free space or vacuum, all electromagnetic waves travel with the speed of light, given by $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$.
• Energy Distribution: The energy in an electromagnetic wave is divided equally between the electric field vector and the magnetic field vector.
• No Charge: These waves are uncharged, hence they are not deflected by electric and magnetic fields.

Electromagnetic Spectrum\nThe orderly distribution of electromagnetic waves in accordance with their wavelength or frequency into distinct groups is called the electromagnetic spectrum:

1. Radio Waves: Used in radio and television communication.
2. Microwaves: Used in radar systems and microwave ovens.
3. Infrared Waves: Used in remote controls and physical therapy.
4. Visible Light: Used to give us the sensation of sight.
5. Ultraviolet Rays: Used in purifying water and detecting forged documents.
6. X-rays: Used in medical diagnostics (fractures).
7. Gamma Rays: Used in cancer treatment and nuclear research.

Numerical Problem Solution

Given:

• Electric field, $E_x = 3.1 \times 10^4 \text{ V/m}$
• Speed of light, $c = 3 \times 10^8 \text{ m/s}$
• Permeability of free space, $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$
• Permittivity of free space, $\epsilon_0 = 8.854 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$

Step 1: Calculate the magnitude of magnetic field ($B_y$)\nUsing the relation between electric and magnetic fields in an electromagnetic wave: $$c = \frac{E_x}{B_y}$$ $$B_y = \frac{E_x}{c}$$ $$B_y = \frac{3.1 \times 10^4 \text{ V/m}}{3 \times 10^8 \text{ m/s}} = 1.033 \times 10^{-4} \text{ T}$|

Step 2: Calculate the total energy density ($u$)\nThe total energy density is the sum of electric energy density ($u_E$) and magnetic energy density ($u_B$). Since $u_E = u_B$, the total energy density is: $$u = 2 u_E = 2 \times \left(\frac{1}{2} \epsilon_0 E^2\right) = \epsilon_0 E^2$$ $$u = (8.854 \times 10^{-12}) \times (3.1 \times 10^4)^2$$ $$u = 8.854 \times 10^{-12} \times 9.61 \times 10^8$$ $$u = 8.509 \times 10^{-3} \text{ J/m}^3$$