Question

\( \text{ The electric field of an EM wave in free space is given by } E = 10 \cos(10^7 t + kx) \, \hat{j} \, \text{V/m, where } t \text{ and } x \text{ are in seconds and meters, respectively. It can be inferred that:}\)
Which of the following pairs of statements are correct:
(A) Wavelength λ is 188.4 m,
(B) Wave number k is 0.33 rad/m,
(C) Wave amplitude is 10 V/m,
(D) Wave is propagating along +x-direction.

• (A) and (C)
• (A) and (B)
• (B) and (C)
• (C) and (D)

Answer 1

The Correct Option is A

Given Wave Equation: 

\( E = 10\cos(10^7t + kx)\hat{j} \) 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.