Infinitesimal Dipole

Outline Maxwell’s equations – Wave equations for A and for  Power: Poynting Vector Dipole antenna

Maxwell Equations Ampere: Faraday: Gauss:

Constitution Relation

Vector Magnetic PotentialA Applying in Faraday’s Law:  is the Electric Scalar Potential

Ampere’s Law:

Lorentz’ condition Assuming The wave equation

Gauss’ Law

Wave equation For sinusoidal fields (harmonics): where

Outline Maxwell’s equations – Wave equations for A and for  Power: Poynting Vector Dipole antenna

Poynting Vector

Average Poynting Vector S In free space:

Outline Maxwell’s equations – Wave equations for A and for  Power: Poynting Vector Infinitesimal Dipole antenna

Find A from Dipole with current J Line charge w/uniform charge density,  L x z  r r 0 JzJz AzAz Assume the simplest solution A z (r):

To find…. Assume the simplest solution A z (r):

Homogenous Equation (J=0) Which has general solution of:

Apply B.C. If radiated wave travels outwards from the source: To find C2, let’s examine what happens near the source. (in that case k tends to 0) So the wave equation reduces to

Now we integrate the volume around the dipole: And using the Divergence Theorem

Comparing both, we get:

Now from A we can find E & H Using the Victoria IDENTITY: And Substitute:

The magnetic filed intensity from the dipole is:

Now the E field:

The electric field from infinitesimal dipole:

field r> 2D 2 / Note that the ratio of E/H is the intrinsic impedance of the medium.

Power :Hertzian Dipole

Radiation Resistance