An air filled coaxial cable designed for maximum power handling has a characteristic impedance of 30. Whereas, an air-filled coaxial cable designed for lower insertion loss or attenuation(maximum efficiency) has characteristic impedance of 70. Hence as a compromise between these two factors, around 50 is chosen as a standard for radio communication.

The impedance of a coaxial cable depends on

(1)

where, diameter of outer conductor

diameter of inner conductor

characteristic impedance of the cable

permmitivity of free-space

relative permittivity of dielectric

From Eq.(1), we can conclude that it is the ratio that defines the characteristic impedance of the cable.

The maximum power handling capacity of air filled coaxial line is set by the dielectric breakdown voltage of air.

For coaxial cable, the electrical field at a distance r along the radial vector is

(2)

The voltage is maximum at the surface of the cable, and is given by

(3)

where,

dielectric voltage breakdown of air in V/m

Peak voltage at the surface of cable

Therefore, the maximum power handling capacity of the cable is

(4)

I don’t think you really answer the question, why 50 Ohms. It is really tied to minimizing losses with a copper-based coax line. Take a peek at

http://www.am1.us/wp-content/Protected_Papers/U11594_Coaxial_Line_and_50_Ohms_v1.pdf

for a derivation that starts very similarly with what you have here. Regards,