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.

Contents

Analytical Solution

wdata/Dipole3.jpg

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:

da2db90788dea6911c220ee88f65338a.png Eqn:7.1
30beaa26fc7503a0fee71eec8961e186.png Eqn:7.2

where,

a7cfe7876dc2dc746bf118b1f5f1e515.png

Retarded Scalar Potential

38d199dd20dead0e9102218693b89eae.png Eqn:7.3

From the Law of Cosines:

fcd51ccdcd69cbc1357de2fbf19da3cc.png Eqn:7.4

We get:

aaf3aada2b1ac88c43abf3987d946253.png Eqn:7.5

and

50ed14c00fc8f60bb05345d1a86a2375.png Eqn:7.6

and given:

7255c5013d24e5eb057044ccbf23efbb.png

we get:

87eb0f4bcbbd2efe40e7718212f96466.png Eqn:7.7

So that:

9dd0481de52c79a598ccdf814cd59a33.png Eqn:7.8

and:

a17b2af6c6b1b7acea1fe3ef859bae3d.png Eqn:7.9

Approximation 1: assume: d << r

Then:

0348cde54a138731892fee67b030286f.png Eqn:7.10
e30449b94788f0149adaacca158e3fb5.png Eqn:7.11

From the 1st order Taylor’s series:

fe48ec5f1209e2f36a5ffb59b8888280.png Eqn:7.12

We get:

7d617844435d0006cd83f406b78c37a0.png Eqn:7.13

and:

1c9037de032432c79f6acd731183232f.png Eqn:7.14

And from the 1st order Taylor’s series:

9b70407d8c154c42acc4ffd217a0629f.png Eqn:7.15

We get:

f4819dceafacfff2675ed96a8b1a3e7a.png Eqn:7.16

and:

5aaab8d963f48089800b64cc7238a947.png Eqn:7.17

Substituting we get:

5cabb67835d28c3d97bcdd65e32bad92.png Eqn:7.18

And:

7dd95bcdec5518ee40feff77d2747d37.png Eqn:7.19

And from the identity:

6386552f75ffffb7bea157d45a1ae1a8.png Eqn:7.20

We get

6c31f6a6145f9013175adfd5d955cd36.png Eqn:7.21

and

481a99b89f587660ebc30f0f72768005.png Eqn:7.22

Approximation 2: assume: d << c / omega, or…

94301bc04821f5c48e0a09af6986843d.png

Also, for small angles:

b1af7726fe2ba4d37923ec8cfcaec36e.png

Using approximation 2, we get

ecb9951e92a37c0146267be15e206b50.png Eqn:7.23

and

b7a796bb3c0bd1027c4998d0693db601.png Eqn:7.24

Substituting,

b076dad88e15292c0c65c88e2e2c1c5c.png Eqn:7.25

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

579d8bf89a9d5a2226a731e038765438.png 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

6ee2c2523629823fecf832863dd8f127.png Eqn:7.27

The current flowing through the dipole wire:

4bf2378ec1c7a4d24fd2e7ec8496988a.png Eqn:7.28

and so…

4fbd45cdf241d317719f1249a1029481.png Eqn:7.29

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

94cca59a0b90e95e44b08e7be10d84b4.png 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.

3dae16091e0ba825593c0c55c4bdf273.png Eqn:7.31
450d59b3824d7dd269fe0b808455eb12.png Eqn:7.32
782a2025031fa6afe5c10cb60adafea0.png 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:

5103532dd995cd7c2716baf656a37e8b.png Eqn:7.34
49f5c6f74bada7c0909bbfd12c9359d6.png Eqn:7.35

Electric Field

Calculating the gradient of V

bbf2dd1a139821f6dbdce15fe3312f5e.png Eqn:7.36

Calculating the partial derivative of A

0d6aff81a8799038cdef27388cdce331.png Eqn:7.37

and from the idenity:

8c65d83de27b5601a5d16560669db004.png

we get…

c85409d1c298f2551a5b3ce5cae2f4dd.png Eqn:7.38

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

cb57704dacc2c794173fe0a143d9a669.png Eqn:7.39

Combining gives:

3dae16091e0ba825593c0c55c4bdf273.png Eqn:7.40
46124764ffbb04b161d1f8b85922b83f.png Eqn:7.41

Reducing terms gives:

59618f629de03dd6ade6d31e38939d11.png Eqn:7.42

Where, k (the wave number) is…

290c001b3b29a226f123121ce1700aff.png

Magnetic Field

Calculate the magnetic flux density vector:

450d59b3824d7dd269fe0b808455eb12.png Eqn:7.43
8cccf4fd02c51608e4fb70eac218db16.png Eqn:7.44
5146cc577769610591d6b7fc4d900f11.png Eqn:7.45

And the magnetic field vector:

782a2025031fa6afe5c10cb60adafea0.png Eqn:7.46
677098ece618207b960c81b9d913a582.png 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:

a1f4410bf10fc0585d3e74a790d2faeb.png Eqn:7.48
19ab4d2b572861f0621e45c6ab867edc.png Eqn:7.49
aeb8a7e4448c6dbecc3ee438615c8590.png Eqn:7.50
649426918fe0bcd3d37493deee94309d.png Eqn:7.51

Magnetic Field:

ffa5f3906df7913336ad6f04f8001123.png Eqn:7.52
eec9326d030f4a7ba31fff9344f5e903.png Eqn:7.53
dc6cd4d90a0a0b9d011e70ae4117df6e.png Eqn:7.54

E,H ready for simulation

And finally, in a form ready for simulation:

16d191d4f591526305319d265f994252.png Eqn:7.55
7d57c1bbd1fbf3a0ca674ca0f119d051.png Eqn:7.56
140c9a10e052658bcefae1b0645ec415.png Eqn:7.57
c2c7307e0b7f0aa801512abc77e034eb.png Eqn:7.58
8c18cd80e904e48c4c884124adea07cd.png Eqn:7.59
3fa64c52204b3e73d0e31f9eef682bb2.png Eqn:7.60
1e0d7b77e2ef27bbb1f0dd51dac49127.png Eqn:7.61
ec4e10b2d5da7a4ba0f11169fe4876fe.png Eqn:7.62

See Also

References

  • GriffithsED – Introduction to ElectroDynamics, (3rd edition) 1999

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