Question

Depict the variation of electric field (\(\vec{E}\)) and magnetic field (\(\vec{B}\)) with respect to the direction of propagation of an electromagnetic wave. Write their two important characteristics.

Hint:

The directional vector equation for any EM wave propagation is always given by the unit vector of the Poynting vector: \(\hat{k}_{\text{prop}} = \hat{E} \times \hat{B}\). Keep in mind that \(\vec{E}\) and \(\vec{B}\) reach their maxima and minima at the exact same positions and times, meaning they are completely in phase.

Answer 1

Concept: An electromagnetic wave is a self-propagating wave composed of oscillating electric and magnetic fields. These fields oscillate in mutually perpendicular planes, and both are simultaneously perpendicular to the direction in which the wave propagates.

Step 1: Spatial depiction of field vectors.
Let us assume that the electromagnetic wave is propagating along the positive X-axis (direction vector \(\hat{i}\)).
• The electric field vector \(\vec{E}\) oscillates sinusoidally parallel to the Y-axis: \[ \vec{E} = E_0 \sin(\omega t - kx) \hat{j} \]
• The magnetic field vector \(\vec{B}\) oscillates sinusoidally in phase with \(\vec{E}\), along the Z-axis: \[ \vec{B} = B_0 \sin(\omega t - kx) \hat{k} \] The cross product \(\vec{E} \times \vec{B}\) points along \(\hat{j} \times \hat{k} = \hat{i}\), which is exactly the direction of wave propagation.

Step 2: Two critical characteristics of Electromagnetic Waves.

• Transverse Nature: Electromagnetic waves are inherently transverse because the oscillations of both the electric field vector \(\vec{E}\) and the magnetic field vector \(\vec{B}\) are strictly perpendicular to the direction of wave travel.
• No Medium Required: They are non-mechanical waves. They can propagate through a vacuum (free space) with a universal constant speed of \(c \approx 3 \times 10^8 \text{ m/s}\), without needing any material medium.