Question

I worked out the fields radiated by an electric dipole in class, and they are also in 11.1.2 of the text. These fields carry energy away and we know the radiated power. From the point of view of the source of the time dependent current (and charge) in the dipole, the antenna could just as well be a resistance with resistance R. Find the value of this resistance, known as the “radiation resistance.”

Griffiths Intro to Electrodynamics

Answer 1

The idea of radiation resistance is like this. As the dipole oscillates in the antenna, then energy is radiated out from its field. If that much of average power is to be radiated as heat, what should be the value of the resistance of the wire connecting the dipoles? That is called radiation resistance. This is different from the electrical resistance in the fact that it has a fictitious behaviour; electrical resistance is always there in the wire, but the radiation resistance will not. Radiation resistance is "felt" when energy is radiated from the dipole as the charged particles lose some energy (in addition to the electrical resistance) in the process of radiation.

The average power radiated out by the dipole antenna is given by

where

Now, for Ohmic conductors, the average power dissipated is given by

Equating (1) and (2), we can define the radiation resistance

Now,

where is the maximum value of charge and is the length of the wire connecting the dipole ends. Also,

Substituting (4) and (5) in (3), we get

Now, for , and since the average value of is 1/2, we get

where

Thus, (7) becomes

Substituting (9) in (6), we get

Now, plugging in the values of the constants, we get the radiation resistance (in Ohms) as

Thus proved.

Now, in the case of an ordinary radio, given that . At a typical broadcasting frequency, for which, the wavelength is of the order of several kilometres, we get

which is negligibly small as compared to the electrical resistance. So, there is no need to worry about the contribution of radiation resistance to the total resistance of the circuit.

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