Question

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

• A. Magnetic field amplitude is 400nT.
• B. Angular frequency of EM wave is π × 10⁸ rad/s
• C. Propagation constant (angular wave number) is 2.1 rad/m
• D. Wavelength of EM wave is 6m.

Answer 1

The Correct Option is C

(A) Magnetic field amplitude is 400nT.

The relationship between the 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 $c$ is the speed of light ($3 \times 10^8\,\text{m/s}$). Substituting the given value of $E_0$:

$B_0 = \frac{120\,\text{NC}^{-1}}{3 \times 10^8\,\text{m/s}} = 40 \times 10^{-8}\,\text{T} = 400 \times 10^{-9}\,\text{T} = 400\,\text{nT}$

This is correct.

(B) Angular frequency of EM wave is $\pi \times 10^8$ rad/s

Angular frequency ($\omega$) is related to frequency ($f$) by:

$\omega = 2\pi f$

$\omega = 2\pi (50 \times 10^6\,\text{Hz}) = \pi \times 10^8\,\text{rad/s}$

This is correct.

(C) Propagation constant (angular wave number) is 2.1 rad/m

The propagation constant ($k$), also known as the angular wave number, is given by:

$k = \frac{2\pi}{\lambda}$

where $\lambda$ is the wavelength. We know that $c = f\lambda$, so $\lambda = \frac{c}{f}$. Thus:

$k = \frac{2\pi f}{c} = \frac{\omega}{c} = \frac{\pi \times 10^8\,\text{rad/s}}{3 \times 10^8\,\text{m/s}} \approx 1.05\,\text{rad/m}$

The given value of $2.1\,\text{rad/m}$ is incorrect.

(D) Wavelength of the EM wave is 6 m.

The wavelength ($\lambda$) is given by:

$\lambda = \frac{c}{f} = \frac{3 \times 10^8\,\text{m/s}}{50 \times 10^6\,\text{Hz}} = 6\,\text{m}$

This is correct.

The correct answer is (C).

Answer 2

Step 1: Calculate Magnetic Field Amplitude ($B_0$)

The magnetic field amplitude ($B_0$) is related to the electric field amplitude ($E_0$) by the speed of light (c):

$B_0 = \frac{E_0}{c}$

Given $E_0 = 120 \ NC^{-1}$ and $c = 3 \times 10^8 \ m/s$.

$B_0 = \frac{120 \ NC^{-1}}{3 \times 10^8 \ m/s} = 40 \times 10^{-8} \ T = 4 \times 10^{-7} \ T = 400 \times 10^{-9} \ T = 400 \ nT$.

The statement "Magnetic field amplitude is 400nT" is correct.

Step 2: Calculate Angular Frequency ($\omega$)

The angular frequency ($\omega$) is related to the frequency (f) by:

$\omega = 2\pi f$

Given frequency f = 50 MHz = $50 \times 10^6 \ Hz$.

$\omega = 2\pi \times 50 \times 10^6 \ rad/s = 100\pi \times 10^6 \ rad/s = \pi \times 10^8 \ rad/s$.

The statement "Angular frequency of EM wave is $\pi \times 10^8$ rad/s" is correct.

Step 3: Calculate Wavelength ($\lambda$)

The wavelength ($\lambda$) is related to the speed of light (c) and frequency (f) by:

$\lambda = \frac{c}{f}$

$\lambda = \frac{3 \times 10^8 \ m/s}{50 \times 10^6 \ Hz} = \frac{300}{50} \ m = 6 \ m$.

The statement "Wavelength of EM wave is 6m" is correct.

Step 4: Calculate Propagation Constant (angular wave number) (k)

The propagation constant (k) is related to the wavelength ($\lambda$) by:

$k = \frac{2\pi}{\lambda}$

Using the calculated wavelength $\lambda = 6 \ m$:

$k = \frac{2\pi}{6 \ m} = \frac{\pi}{3} \ rad/m$

Using the approximate value of $\pi \approx 3.14$:

$k \approx \frac{3.14}{3} \ rad/m \approx 1.047 \ rad/m$

The given propagation constant is 2.1 rad/m. This value is significantly different from our calculated value of approximately 1.047 rad/m.

Therefore, the incorrectly computed value is the propagation constant.

Final Answer: The final answer is Propagation constant (angular wave number) is 2.1 rad/m