We know that the Curl of the magnetic field (H) is equal to the sum of current density (J) and the rate of change of the electric filled (E):

(4.2.1) 
Since the Hertzian Dipole is a conductor, we need only concern ourselves with the current density (J) when calculating the vector potential (A). Integrating current density (J) over the volume of the antenna, is equivalent to integrating current along the length of the antenna (L).
We know that Maxwell's Equations can be solved for single frequency (monochromatic) fields, so we will excite our antenna with a single frequency current:

(4.2.2) 
We can simplify the integral for the vector potential (A) by recognising that:
1. 
Our observation point (P) will be a long way from the antenna and so (r) will be very large.

2. 
The length of the antenna (L) will be very small and so (r') will be very small.

Since r>>r', we can substitute rr' with r.
Because we have decided that the observation point at r will be a long way from the antenna, we must allow for the fact that the observed antenna current will be delayed. The delay will be equal to the distance from the antenna to the observation point rr' (which we have simplified to r), divided by the speed of light (c). The time delay will therefore be approximately equal to r/c and so the observed antenna current becomes:

(4.2.3) 
Since the length, L of the antenna will be very small, we can assume that the current is in phase at all points along its length. Working in the Cartesian coordinate system, the final integral for the vector potential for the magnetic field is therefore:

(4.2.4) 
We will now convert to the spherical coordinate system, which is more convenient when working with radio antenna radiation patterns:
The radial component of the observed current (and therefore vector potential), will be at a maximum when the observer is on the zaxis (that is when θ=0 or θ=π) and will be zero when the observer is in the xyplane:

(4.2.5) 
The angular component of the observed current (and therefore vector potential), in the θ direction will be zero when the observer is on the zaxis (that is when θ=0 or θ=π) and will be at a maximum when the observer is in the xyplane:

(4.2.6) 
Since the observed current (and therefore vector potential) flows along the zaxis, there will be no variation in the ϕ direction. That is to say, that varying ϕ will have no impact on the observed vector potential.

(4.2.7) 
And so the vector potential for the magnetic field (H) expressed using spherical coordinate system is:

(4.2.8) 
