Question

Suppose that the electric field amplitude of electromagnetic wave is $E _{0}=120\, NC ^{-1}$ and its frequency if $f =50 \,MHz$. Then which of the following value incorrectly computed ?

• A. Magnetic field amplitude is $400 \,nT$.
• B. Angular frequency of EM wave is $\pi \times 10^{8} rad / s$
• C. Propagation constant (angular wave number) is $2.1\, rad / m$
• D. Wavelength of EM wave is $6\, m$.

Answer 1

The Correct Option is C

To determine which of the given values is incorrectly computed, we need to calculate each related parameter of the electromagnetic wave using the given data:

• Electric Field Amplitude, $E_0 = 120 \, \text{N/C}$
• Frequency, $f = 50 \, \text{MHz} = 50 \times 10^6 \, \text{Hz}$

Let's analyze each option:

1. Magnetic Field Amplitude:
   The relation between electric field amplitude $E_0$ and magnetic field amplitude $B_0$ in an electromagnetic wave is given by: $$B_0 = \frac{E_0}{c},$$ where $c = 3 \times 10^8 \, \text{m/s}$ is the speed of light.
   Substituting the values, $$B_0 = \frac{120 \, \text{N/C}}{3 \times 10^8 \, \text{m/s}} = 4 \times 10^{-7} \, \text{T} = 400 \, \text{nT}.$$
   This matches the given value, so it is correctly computed.
2. Angular Frequency:
   Angular frequency $(\omega)$ is calculated as: $$\omega = 2\pi f.$$
   For $f = 50 \times 10^6 \, \text{Hz},$ $$\omega = 2\pi \times 50 \times 10^6 = 100\pi \times 10^6 \, \text{rad/s}.$$
   The given value is $\pi \times 10^8 \, \text{rad/s}$, which simplifies to $100\pi \times 10^6 \, \text{rad/s},$ so it is correctly computed.
3. Propagation Constant:
   The propagation constant or angular wave number $(k)$ is given by: $$k = \frac{2\pi}{\lambda},$$ where $\lambda$ is the wavelength. Wavelength $\lambda$ is calculated as: $$\lambda = \frac{c}{f} = \frac{3 \times 10^8}{50 \times 10^6} = 6 \, \text{m}.$$
   Therefore, $$k = \frac{2\pi}{6} = \frac{\pi}{3} \, \text{rad/m} \approx 1.05 \, \text{rad/m}.$$
   The given value is $2.1 \, \text{rad/m}$, which is incorrect.
4. Wavelength of EM Wave:
   We already calculated in option 3 that $\lambda = 6 \, \text{m},$ which matches the given value. Hence, it is correctly computed.

Conclusion: The incorrectly computed value is the Propagation constant (angular wave number) as $2.1 \, \text{rad/m}$.