Understanding the Concept:
The far-field electric field component ($E_\theta$) radiated by an ideal, infinitesimally small Hertzian dipole or a finite-length half-wave dipole oriented vertically along the $z$-axis contains a directional dependency term proportional to:
$$E_\theta \propto \sin\theta$$
Where $\theta$ represents the polar angle measured relative to the longitudinal physical axis of the dipole element.
Step-by-step Structural Breakdown:
• Let us evaluate the radiation pattern behavior at different observation angles ($\theta$):
• When $\theta = 0^\circ$ or $180^\circ$ (directly along the conductor wire line axis):
$$\sin(0^\circ) = 0$$
This confirms that the radiated energy falls to absolute zero along the axis of the dipole wire.
• When $\theta = 90^\circ$ (perpendicular to the dipole wire axis, in the broadside azimuthal plane):
$$\sin(90^\circ) = 1 \quad \text{(Maximum value)}$$
This confirms that the antenna achieves its peak radiation efficiency perpendicular to its structure.
• Mapping this $\sin\theta$ configuration across full 3D space produces an omnidirectional toroid profile, commonly described as a doughnut-shaped geometry.