Question

If the amplitude of electric field vector and frequency of an electromagnetic wave travelling along $z$-direction in vacuum are 180 N C$^{-1}$ and 60 GHz respectively, then the equation of the magnitude of the electric field of the plane electromagnetic wave is

• A. \( E = 180 \sin\left( \pi \left( 1000z - 60 \times 10^9 t \right) \right)~\text{N C}^{-1} \)
• B. \( E = 180 \sin\left( 2\pi \left( 200z - 6 \times 10^{10} t \right) \right)~\text{N C}^{-1} \)
• C. \( E = 180 \sin\left( 2\pi \left( 100z - 3 \times 10^{10} t \right) \right)~\text{N C}^{-1} \)
• D. \( E = 180 \sin\left( 2\pi \left( 10z - 3 \times 10^5 t \right) \right)~\text{N C}^{-1} \)

Hint:

Use the wave equation \( E = E_0 \sin(2\pi(kz - ft)) \) for electromagnetic waves in vacuum. Compute $k$ using \( k = \frac{f}{c} \).

Answer 1

The Correct Option is B

The general form of the electric field of an EM wave traveling in the $z$-direction is: \[ E = E_0 \sin(2\pi (kz - ft)) \] Given: \[ E_0 = 180~\text{N C}^{-1},\quad f = 60~\text{GHz} = 6 \times 10^{10}~\text{Hz} \] The wave number $k = \frac{f}{c} = \frac{6 \times 10^{10}}{3 \times 10^8} = 200~\text{m}^{-1}$ Thus, \[ E = 180 \sin\left( 2\pi (200z - 6 \times 10^{10} t) \right)~\text{N C}^{-1} \]