# Dipole antenna radiation pattern derivation

For a dipole antenna centered on the origin and oriented along the z axis with length L, the far-field radiation pattern can be derived as follows:

The current on the antenna will be approximately sinusoidal, with zeros at the ends of the antenna, represented by

where

Use the current distribution to find the Z component of the vector potential

where

and the magnitude of R can be approximated as

Substituting yields

As , can be neglected in the denominator. However, it cannot be neglected in the exponential as it is a phase offset.

rearranging yields

Eliminate the absolute value by splitting into two integrals:

Flip limits on the first integral and combine

Substitute

Now, integrate it, setting aside the constants and limits temporarily:

Integrate by parts (tan cow):

And again:

Time for some algebra:

so

Now substitute

and plug in the limits and and bring back the constants

simplify

Now, convert to :

This equation for is the general form for the theta component in spherical coordinates of the far-field E field of a dipole antenna of any length oriented along the z-axis. The r component is zero due to the far-field assumption and the phi component is zero due to the electric field's orientation along the z-axis.