Correct Option: (A) $2.0 \times 10^8 \text{ m s}^{-1}$
Step 1 (Formula / Governing Law & Given Data): [0.25 Mark]
The standard wave equation for a magnetic field is given by:
$$B = B_0 \sin(\omega t - kx)$$Comparing this with the given equation $B = 5 \times 10^{-8} \sin(3 \times 10^{10} t - 150 x)$, we identify the parameters:
- Angular frequency, $\omega = 3 \times 10^{10} \text{ rad s}^{-1}$
- Wave number, $k = 150 \text{ rad m}^{-1}$
The velocity of the wave ($v$) is defined by the relation:
$$v = \frac{\omega}{k}$$Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]
Substitute the values of $\omega$ and $k$ into the velocity formula:
$$v = \frac{3 \times 10^{10}}{150}$$Simplify the expression:
$$v = \frac{3}{150} \times 10^{10}$$ $$v = \frac{1}{50} \times 10^{10}$$ $$v = 0.02 \times 10^{10} \text{ m s}^{-1}$$Step 3 (Final Concluded Answer): [0.25 Mark]
Expressing the result in standard scientific notation:
$$v = 2.0 \times 10^8 \text{ m s}^{-1}$$Thus, the velocity of the wave is $2.0 \times 10^8 \text{ m s}^{-1}$, which corresponds to option (A).
💡 EL10 Concept Insight: Think of the wave equation like a speed limit sign; the number attached to time ($t$) tells you how fast the wave oscillates, and the number attached to distance ($x$) tells you how cramped the waves are. By dividing how fast it wiggles by how cramped it is, you get the actual speed at which the wave travels through space!