Electromagnetic Fields For Hertzian Dipoles

This Demonstration shows the electromagnetic fields for an electric dipole or a Hertzian dipole, the electric and magnetic fields, the associated energy densities, and the Poynting vector distributions. You can vary the dipole moment, frequency, and time for either a DC or static dipole field.

THINGS TO TRY

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DETAILS

Snapshot 1: a magnetic field distribution

Snapshot 2: an energy density and Poynting vector distributions

Snapshot 3: a DC electric field

The electromagnetic fields of a Hertzian dipole can be analyzed using the electrical Hertz vector:

Π

e

p

0

4πr

ϵ

0

where is the dipole moment (a vector in the direction in this analysis), and is the distance to the observation point. The energy density is

p

0

z

r

W

W=

1

2

μ

0

2

H

The electric and magnetic fields can be calculated as follows:

E

=∇∇Π

e

H

=ϵ

0

∂

Π

e

∂t

jω∇

ϵ

0

Π

e

j=

-1

In the sinusoidally oscillating dipole of =0,0, in cylindrical coordinates , the field vectors in the - plane are

p

0

p

0

jωt

e

(r,θ,z)

r

z

E

=(E

r

E

z

H

=(0,H

θ

The total energy density and Poynting vector are given by

W

S

W=+

ϵ

0

2

2

E

μ

0

2

2

H

S

=E

H

In the graphics, the field strengths are shown by color and the directions are shown by arrows.

The DC fields are obtained from the above equation, with the magnetic field equal to zero.

Reference

[1] J. A. Stratton,

Electromagnetic Theory

, New York: McGraw-Hill, 1941.RELATED LINKS

PERMANENT CITATION

"" Published: September 4, 2012