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

Provided Wave Equation:

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

Evaluation of Statements:

(A) Wavelength λ = 188.4 m:

• From the wave equation, the angular frequency ω is 10⁷ rad/s.
• The speed of light in free space, c, is 3 × 10⁸ m/s.
• The wavelength is calculated as λ = 2πc/ω = (2π × 3 × 10⁸)/10⁷ ≈ 188.4 m.
• This statement is correct.

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

• The wave number k = ω/c = 10⁷/(3 × 10⁸) ≈ 0.033 rad/m.
• The provided value (0.33) is 10 times larger than the calculated value (0.033).
• This statement is incorrect.

(C) Wave amplitude = 10 V/m:

• The wave amplitude is directly observable as the coefficient 10 in the equation.
• This statement is correct.

(D) Propagation along +x direction:

• The phase term is (ωt + kx). A positive sign between ωt and kx indicates propagation in the -x direction.
• Propagation in the +x direction would be represented by (ωt - kx).
• This statement is incorrect.

Correct Statements:

Statements (A) and (C) are the only correct ones.