# Analytic Solution: Hertzian Dipole Sine Wave Source

The analytical solution to the Hertzian Dipole (with a Sine Wave source) problem for comparison with the FDTD simulation.

## Analytical Solution A Hertzian Dipole consisting of two points of charge +q and -q centered around the origin and separated by distance d and observed at a point r (r >> d). From Griffiths pp. 444-448

The Formula for a Hertzian Dipole with a sinusoidal oscillation of frequency omega is: Eqn:7.1 Eqn:7.2

where, ### Retarded Scalar Potential Eqn:7.3

From the Law of Cosines: Eqn:7.4

We get: Eqn:7.5

and Eqn:7.6

and given: we get: Eqn:7.7

So that: Eqn:7.8

and: Eqn:7.9

Approximation 1: assume: d << r

Then: Eqn:7.10 Eqn:7.11

From the 1st order Taylor’s series: Eqn:7.12

We get: Eqn:7.13

and: Eqn:7.14

And from the 1st order Taylor’s series: Eqn:7.15

We get: Eqn:7.16

and: Eqn:7.17

Substituting we get: Eqn:7.18

And: Eqn:7.19

And from the identity: Eqn:7.20

We get Eqn:7.21

and Eqn:7.22

Approximation 2: assume: d << c / omega, or… Also, for small angles: Using approximation 2, we get Eqn:7.23

and Eqn:7.24

Substituting, Eqn:7.25

factoring and reducing terms gives (Griffiths page446, equation 11.12): Eqn:7.26

The first term is the far field, the second term is the near field.

### Retarded Vector Potential

Griffiths page 423, equation 10.19 retarded potentials Eqn:7.27

The current flowing through the dipole wire: Eqn:7.28

and so… Eqn:7.29

Approximation: use value at center of dipole, to get (Griffiths page446, equation 11.17): Eqn:7.30

### Electric and Magnetic Fields

Now that the scalar and vector potentials have been determined, the electric and magnetic fields can be computed. Eqn:7.31 Eqn:7.32 Eqn:7.33

Using the formulas for gradient and curl in spherical coordinates
Equation B.10 and Equation B.11 and noting that neither A or V have any terms dependent on phi, we have: Eqn:7.34 Eqn:7.35

#### Electric Field Eqn:7.36

Calculating the partial derivative of A Eqn:7.37

and from the idenity: we get… Eqn:7.38

And from the conversion Equation B.6 between Z and R,Theta, we have: Eqn:7.39

Combining gives: Eqn:7.40 Eqn:7.41

Reducing terms gives: Eqn:7.42

Where, k (the wave number) is… #### Magnetic Field

Calculate the magnetic flux density vector: Eqn:7.43 Eqn:7.44 Eqn:7.45

And the magnetic field vector: Eqn:7.46 Eqn:7.47

### Electric and Magnetic Fields in Cartesian coordinates

For comparison with the results from the fdtd simulation, the fields must be converted
to Cartesian coordinates

Using the conversions from R,Theta,Phi to X,Y,Z, Equation B.7, Equation B.8, Equation B.9 we have:

#### Electric Field: Eqn:7.48 Eqn:7.49 Eqn:7.50 Eqn:7.51

#### Magnetic Field: Eqn:7.52 Eqn:7.53 Eqn:7.54

And finally, in a form ready for simulation: Eqn:7.55 Eqn:7.56 Eqn:7.57 Eqn:7.58 Eqn:7.59 Eqn:7.60 Eqn:7.61 Eqn:7.62