Elementary doublet (infinitesimal dipole) – What is its directive gain? For an ideal very short dipole (elementary doublet) in free space, identify the correct value of directive gain.

Introduction:
Directive gain (directivity) quantifies how concentrated an antenna's radiation is in the strongest direction relative to an isotropic radiator. The elementary doublet (infinitesimal dipole) is a theoretical reference antenna used in classical electromagnetics to derive radiation fields and patterns and to benchmark other element designs.

Given Data / Assumptions:

• Electrically short (length ≪ λ) center-fed dipole in free space.
• Sinusoidal current distribution approximated as uniform for the infinitesimal limit.
• Lossless antenna, so gain equals directivity for this idealized case.

Concept / Approach:

The elementary doublet has a doughnut-shaped power pattern with maximum radiation in the plane perpendicular to the dipole and nulls along the dipole axis. Integrating the normalized power pattern over 4π steradians yields a directivity D0 = 1.5 (that is, 1.76 dBi). This serves as a baseline for understanding how longer dipoles, arrays, or apertures increase directivity beyond this value by narrowing the main lobe and reducing beamwidth.

Step-by-Step Solution:

Verification / Alternative check:

Textbook tables list D0(short dipole) = 1.5 and D0(half-wave dipole) ≈ 1.64, confirming the established benchmarks and illustrating the modest increase in directivity as the electrical length grows to λ/2.

Why Other Options Are Wrong:

Values like 10 or 100 indicate highly directive arrays or large apertures, not a single short dipole. 0.5 is less than isotropic (not possible for directivity). 3 corresponds to a higher directivity than even a half-wave dipole and is inconsistent with the elementary doublet model.

Common Pitfalls:

Confusing directivity with gain in dBi (a logarithmic unit); mixing up the elementary doublet with the half-wave dipole values; forgetting that real efficiency less than 100% lowers realized gain but not the theoretical directivity number.

Final Answer:

1.5.