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Step-by-Step Solution
Step 1: Identify the Key Parameters
• Wavelength of the electromagnetic wave, $ \lambda = 8\,\text{mm} = 8 \times 10^{-3}\,\text{m}.$
• The electric field ($E$) is given to have a maximum magnitude (amplitude) of $60\,\text{V/m}$ in the $y$-direction.
• The wave is propagating along the $x$-direction in vacuum, where the speed of light is $ c = 3 \times 10^8\,\text{m/s}.$
Step 2: Relate Wavelength to Wave Number
The wave number $k$ is given by
$$ k = \frac{2\pi}{\lambda}. $$
Substituting $ \lambda = 8 \times 10^{-3} \,\text{m} $:
$$ k = \frac{2\pi}{8 \times 10^{-3}} = \frac{\pi}{4} \times 10^3\,\text{m}^{-1}. $$
Step 3: Determine the Angular Frequency
The angular frequency $ \omega $ in vacuum is related to $k$ by
$$ \omega = c\,k. $$
Hence,
$$ \omega = 3 \times 10^8 \times \left( \frac{\pi}{4} \times 10^3 \right) = \frac{\pi}{4} \times 3 \times 10^{11} \,\text{rad/s}. $$
Step 4: Use the Relationship between Electric and Magnetic Field Amplitudes
For an electromagnetic wave in vacuum:
$$ E_0 = c\,B_0. $$
Given $ E_0 = 60\,\text{V/m} $ and $ c = 3 \times 10^8\,\text{m/s} $, we find:
$$ B_0 = \frac{E_0}{c} = \frac{60}{3 \times 10^8} = 2 \times 10^{-7}\,\text{T}. $$
Step 5: Write the Correct Wave Equations
The general form for the electric and magnetic fields of a plane electromagnetic wave propagating along $x$ is:
$$ E_y(x,t) = E_0 \sin\bigl[k(x - ct)\bigr]\,\hat{j}, $$
$$ B_z(x,t) = B_0 \sin\bigl[k(x - ct)\bigr]\,\hat{k}. $$
Substituting $ E_0 = 60\,\text{V/m} $, $ B_0 = 2 \times 10^{-7}\,\text{T} $, and $ k = \frac{\pi}{4} \times 10^3\,\text{m}^{-1}$ into the argument $ k(x - ct) = \left(\frac{\pi}{4} \times 10^3\right)(x - 3 \times 10^8 t), $ we obtain:
$ E_y = 60 \sin\Bigl[\frac{\pi}{4} \times 10^3\,\bigl(x - 3 \times 10^8 t\bigr)\Bigr] \hat{j} \,\text{V/m}, $
$ B_z = 2 \times 10^{-7} \sin\Bigl[\frac{\pi}{4} \times 10^3\,\bigl(x - 3 \times 10^8 t\bigr)\Bigr] \hat{k} \,\text{T}. $
These match the correct choice given in the problem statement.
Step 6: Verify Consistency with the Speed of Light
• The phase term $(x - 3 \times 10^8 t)$ confirms that the wave is traveling in the $+x$ direction at speed $3 \times 10^8\,\text{m/s}$.
• The ratio $\frac{E_0}{B_0} = 3 \times 10^8\,\text{m/s}$ confirms the wave is an electromagnetic wave in vacuum.
Hence, the correct equations for the electric and magnetic fields are:
$ E_y = 60 \sin\Bigl[\frac{\pi}{4} \times 10^3(x - 3 \times 10^8 t)\Bigr]\hat{j}\,\text{V/m}, $
$ B_z = 2 \times 10^{-7} \sin\Bigl[\frac{\pi}{4} \times 10^3(x - 3 \times 10^8 t)\Bigr]\hat{k}\,\text{T}. $
