Know the two numbers and which is which. For a half-wave dipole the directivity as a ratio is

\(D=1.64\)

and the question asks for it in decibels:

\(D_{dB}=10\log_{10}(1.64)=10\times0.2148\)

\(D_{dB}=2.15\ \text{dB}\)

which is option 3.

The trap is option 4. 1.64 is the correct directivity, but as a plain ratio, not in decibels. The question is testing whether the two forms are kept apart — and 1.64 dB would actually correspond to a ratio of about 1.46, not 1.64. Whenever both a number and its decibel value appear among the choices, check which the question asked for.

Option 1 is the other classic distractor. 1.76 dB is the directivity of the Hertzian (infinitesimal) dipole, whose ratio is 1.5:

\(10\log_{10}(1.5)=1.76\ \text{dB}\)

The half-wave dipole is slightly more directive than the elementary one because its current distribution is sinusoidal rather than uniform, which narrows the beam a little — from a half-power beamwidth of 90° down to 78°.

Where 1.64 comes from. The far field of a half-wave dipole is

\(E_\theta \propto \dfrac{\cos\left(\dfrac{\pi}{2}\cos\theta\right)}{\sin\theta}\)

Integrating \(|E_\theta|^{2}\) over the sphere gives a radiation resistance of 73 Ω and, on forming \(D=4\pi U_{max}/P_{rad}\), the value 1.64.

Why 2.15 dB is worth memorising above all. It is the conversion between the two gain references:

\(G_{dBi}=G_{dBd}+2.15\)

Manufacturers quote antenna gain sometimes against isotropic (dBi) and sometimes against a half-wave dipole (dBd), and 2.15 dB is the constant that reconciles them.

Summary of the standard values. Half-wave dipole: D = 1.64 = 2.15 dBi, R_rad = 73 Ω, length λ/2. Hertzian dipole: D = 1.5 = 1.76 dBi.

Hence, the directivity of a half-wave dipole is 2.15 dB.