For a very short doublet (electrically short dipole) of length l ≪ λ, the radiation resistance varies with which power of the electrical length? (Let β = 2π/λ; electrical length is β l.)

Introduction / Context:
The radiation resistance R_r of an antenna models how effectively it converts current into radiated power. For electrically short dipoles (length much smaller than wavelength), R_r is small and depends strongly on the electrical length.

Given Data / Assumptions:

• Short dipole (uniform current approximation), l ≪ λ.
• Electrical length β l where β = 2π/λ.
• Free-space operation for simplicity.

Concept / Approach:

Classical antenna theory gives R_r ≈ 80 * π^2 * (l/λ)^2 for a short dipole. Since β = 2π/λ, (l/λ)^2 ∝ (β l)^2. Therefore, radiation resistance is proportional to the square of the electrical length, reflecting the weak radiation of very small antennas.

Step-by-Step Solution:

Verification / Alternative check:

Using the approximate constant: R_r ≈ 80π^2(l/λ)^2 confirms the quadratic dependence and yields numerical values that match measurements for l ≪ λ.

Why Other Options Are Wrong:

• Linear or square-root dependence and cubic dependence do not match the established short-dipole formula.
• Independence of β l ignores the small-antenna limit behavior.

Common Pitfalls:

Confusing radiation resistance with input resistance including loss; applying the formula outside the l ≪ λ range.

Final Answer:

Proportional to (β l)^2