Question

The expression of magnetic fields associated with four electromagnetic waves are given below :
I. $B_1 = (4 \times 10^{-6} \text{ T}) \sin [0.7 \times 10^3 x + 1.4 \times 10^{11} t]$
II. $B_2 = (2 \times 10^{-7} \text{ T}) \sin [0.6 \times 10^3 x + 1.5 \times 10^{11} t]$
III. $B_3 = (3 \times 10^{-5} \text{ T}) \sin [0.5 \times 10^3 x + 1.5 \times 10^{11} t]$
IV. $B_4 = (5 \times 10^{-4} \text{ T}) \sin [0.2 \times 10^4 x + 4.8 \times 10^{11} t]$
Which wave is travelling in free space ?

• A. I
• B. II
• C. III
• D. IV

Hint:

When visually inspecting wave equations, quickly glancing at the ratio of the coefficients of '$t$' over '$x$' immediately reveals the wave's phase velocity.

Answer 1

The Correct Option is C

Concept:
• The mathematical representation of a propagating electromagnetic wave involves a sinusoidal function typically in the standard form $B = B_0 \sin(kx \pm \omega t)$.

• Here, $k$ definitively represents the angular wave number, and $\omega$ represents the angular frequency.

• The physical propagation speed $v$ of any such wave is robustly determined by the mathematical ratio of its angular frequency to its wave number: $v = \frac{\omega}{k}$.

• For an electromagnetic wave strictly travelling through the vacuum of free space, this calculated speed must exactly equal the universal speed of light, $c \approx 3 \times 10^8 \text{ m/s}$.

Step 1: Extract parameters and calculate speed for Wave I
From the given expression for $B_1$, we identify the coefficients:
Angular wave number, $k_1 = 0.7 \times 10^3 \text{ rad/m}$.
Angular frequency, $\omega_1 = 1.4 \times 10^{11} \text{ rad/s}$.
Calculate the corresponding wave speed $v_1$:
\[ v_1 = \frac{\omega_1}{k_1} = \frac{1.4 \times 10^{11}}{0.7 \times 10^3} \]
\[ v_1 = 2 \times 10^8 \text{ m/s} \]
This speed is significantly less than $c$, meaning it is traveling in a denser medium, not free space.

Step 2: Extract parameters and calculate speed for Wave II
From the given expression for $B_2$:
$k_2 = 0.6 \times 10^3 \text{ rad/m}$.
$\omega_2 = 1.5 \times 10^{11} \text{ rad/s}$.
Calculate the corresponding wave speed $v_2$:
\[ v_2 = \frac{\omega_2}{k_2} = \frac{1.5 \times 10^{11}}{0.6 \times 10^3} \]
\[ v_2 = 2.5 \times 10^8 \text{ m/s} \]
This is also clearly not equal to the definitive speed of light $c$.

Step 3: Extract parameters and calculate speed for Wave III
From the given expression for $B_3$:
$k_3 = 0.5 \times 10^3 \text{ rad/m}$.
$\omega_3 = 1.5 \times 10^{11} \text{ rad/s}$.
Calculate the corresponding wave speed $v_3$:
\[ v_3 = \frac{\omega_3}{k_3} = \frac{1.5 \times 10^{11}}{0.5 \times 10^3} \]
\[ v_3 = 3.0 \times 10^8 \text{ m/s} \]
This meticulously matches the widely accepted speed of light in free space perfectly.

Step 4: Extract parameters and calculate speed for Wave IV for completeness
From the given expression for $B_4$:
$k_4 = 0.2 \times 10^4 \text{ rad/m}$.
$\omega_4 = 4.8 \times 10^{11} \text{ rad/s}$.
Calculate the corresponding wave speed $v_4$:
\[ v_4 = \frac{\omega_4}{k_4} = \frac{4.8 \times 10^{11}}{0.2 \times 10^4} = 24 \times 10^7 = 2.4 \times 10^8 \text{ m/s} \]
This is also incorrect for free space.

Step 5: Conclusion
Only Wave III possesses a calculated propagation velocity that exactly equals $3 \times 10^8 \text{ m/s}$. Therefore, only Wave III is actively travelling in free space. This solidly aligns with option (C).