Question 8.11: Find the directivity of the Hertzian dipole.

Use the definition (8.43) for the directivity, including the normalized radiation intensity for a short dipole I _{n}(θ, \phi ) = \sin ^{2} θ, and obtain

D = \frac{I(\theta , \phi )_{max}}{\left\lgroup\frac{P_{rad}}{4\pi } \right\rgroup } = \frac{4\pi }{\oint_{4\pi }I_{n} (\theta , \phi ) d\Omega } = \frac{4\pi }{\Omega _{A}}      (8.43)

For the short dipole, the directivity is D = 1.5 or 10 \log_{10}( 1.5) = 1.76  dB. A more complicated calculation performed for the half-wave dipole leads to a slightly higher value for the directivity D = 1.64 or 2.15 dB as given in Problem 8.4.6