Electromagnetic Waves

In an electromagnetic wave, both electric and magnetic fields contribute to the energy density of the wave. The energy density associated with the electric field and the magnetic field can be expressed as:

Here, is the permittivity of free space, is the permeability of free space, is the electric field strength, and is the magnetic field strength. In free space, the relationship between the magnitudes of the electric field and magnetic field in an electromagnetic wave is given by the equation , where is the speed of light in a vacuum.

Substituting into the expression for , we have:

Now, the speed of light can also be expressed as . Substituting this into the expression for :

This shows that the energy density of the electric field is equal to the energy density of the magnetic field, resulting in the ratio .

Thus, the correct answer is 1:1.

To solve this problem, we need to correctly match the electromagnetic (EM) waves listed in Column-1 with their corresponding uses in Column-2. Below is a structured approach to find the correct matches:

Here's the reasoning for each match:

• (A) Radio waves are used for broadcasting because they can carry signals over long distances without being absorbed by the atmosphere.
• (B) Microwaves are used for cooking, primarily in microwave ovens, as they can excite water molecules and heat food quickly.
• (C) Infrared is used in night vision equipment, as it can detect heat emitted by objects in low-light conditions.
• (D) X-rays are used in medical imaging due to their ability to pass through soft tissues while being absorbed by denser materials like bones, providing clear contrast images.

Thus, the correct matching is: (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

The electromagnetic spectrum is a range of all types of electromagnetic radiation, which vary in wavelength and frequency. To arrange different types of waves in decreasing order of wavelength, we follow this sequence:

• Radio waves: Typically have the longest wavelengths in the electromagnetic spectrum, ranging from about 1 millimeter to 100 kilometers or more.
• Microwaves: Have shorter wavelengths than radio waves, generally from 1 millimeter to 1 meter.
• Infrared waves: These range from about 700 nanometers to 1 millimeter.
• Visible waves: The part of the electromagnetic spectrum that human eyes can see, with wavelengths from approximately 380 nanometers to 750 nanometers.
• X-rays: Have much shorter wavelengths, ranging from 0.01 to 10 nanometers.

Arranging the options in the decreasing order of wavelength, the correct sequence is: Radio waves, microwaves, Infrared waves, visible waves, X-rays

Given Wave Equation:

V/m

Analysis of Each Statement:

(A) Wavelength λ = 188.4 m:

• From the wave equation, angular frequency ω = 10⁷ rad/s
• In free space, speed of light c = 3 × 10⁸ m/s
• Wavelength λ = 2πc/ω = (2π × 3 × 10⁸)/10⁷ ≈ 188.4 m
• Correct

(B) Wave number k = 0.33 rad/m:

• Wave number k = ω/c = 10⁷/(3 × 10⁸) ≈ 0.033 rad/m
• Given value is 10× larger (0.33 vs 0.033)
• Incorrect

(C) Wave amplitude = 10 V/m:

• Directly visible from the equation coefficient (10)
• Correct

(D) Propagation along +x direction:

• Phase is (ωt + kx) → indicates -x direction propagation
• Incorrect (would need (ωt - kx) for +x)

Correct Statements:

Only (A) and (C) are correct.