(a) Variation of Fields and Characteristics

Step 1 (Depiction and Characteristics): [1 Mark]

Electromagnetic waves are transverse in nature. The electric field ($\vec{E}$) and magnetic field ($\vec{B}$) oscillate sinusoidally in mutually perpendicular directions, both being perpendicular to the direction of wave propagation.

Diagrammatic Representation:

If the wave propagates along the $x$-axis, the fields are represented as:

$$E_y = E_0 \sin(kx - \omega t)$$ $$B_z = B_0 \sin(kx - \omega t)$$

Two Important Characteristics:

• Electromagnetic waves do not require any material medium for propagation; they can travel through a vacuum.
• The ratio of the amplitudes of the electric and magnetic fields is equal to the speed of light in vacuum: $\frac{E_0}{B_0} = c$.

Step 2 (Derivation of Velocity): [1.5 Marks]

From Maxwell's equations for free space, the wave equations for $\vec{E}$ and $\vec{B}$ are given by:

$$\frac{\partial^2 E}{\partial x^2} = \mu_0 \varepsilon_0 \frac{\partial^2 E}{\partial t^2}$$

Comparing this with the standard wave equation $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$, we identify:

$$\frac{1}{v^2} = \mu_0 \varepsilon_0$$ $$v^2 = \frac{1}{\mu_0 \varepsilon_0}$$

Step 3 (Final Concluded Answer): [0.5 Mark]

Taking the square root on both sides, we obtain the velocity of the electromagnetic wave in free space:

$$v = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$$

Substituting the values $\mu_0 = 4\pi \times 10^{-7} \, \text{T m/A}$ and $\varepsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{N m}^2$, we get $v \approx 3 \times 10^8 \, \text{m/s}$, which is the speed of light ($c$).

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💡 EL10 Concept Insight: Think of an electromagnetic wave like a synchronized dance where the electric field wiggles up and down and the magnetic field wiggles side to side; because they are constantly creating each other through their changes, they push the wave forward through space at the fastest speed possible in the universe.