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Pd = ( Pt * Gt) / 4 * Pi * r^2 where Pt (in Watts) is the power transmitted from the antenna.

All these variables are scalar values so if there were multiple antennas and we wanted to calculate the total power density due to all of them at a given point, you could calculate the individual power density due to each one and add them together.

Now, the power density in the far field is also given by:

Pd = Erms^2 / 377

where Erms is the RMS electric field (in V/m) at the location and 377 ohms is the intrinsic impedance of free space.

Having already calculated the total power density, we can substitute it into the above equation and find a value for the total RMS electric field due to all the sources at the point of interest.

This is where things seem to go wrong.

The electric field is a vector quantity and so when multiple fields meet, they have to be added using vector addition. You can't add two RMS electric field values together unless they are totally in phase. However, using the above methodology, we seemed to have arrived at a value for the total RMS electric field. It would seem the answer has to be wrong because we haven't taken the phase differences into account. But since these equations are scalar, where do you do that?

Does anyone have any suggestions?