====== Radiation resistance ======
Ohm's definition of [[resistance]] is given as $R=V/I$, so resistance is a ratio of the voltage and current. So the power of an Ohmic circuit

$$ P = VI = I^2R = \frac{V^2}{R} $$

and in case of alternating current, we have to average over time to get the power which, for sinusoidal currents, become

$$ \langle P\rangle = \langle I^2\rangle R = \frac{I_0^2R}{2} $$

because for sinusoidal currents, $\langle I^2\rangle = I_0^2/2$ which gives the definition of the **radiation resistance** of an antenna

$$ R \equiv \frac{2\langle P\rangle}{I_0^2} = \frac{2\pi^2}{3c} \left(\frac{l}{\lambda}\right)^2 $$

where the time-averaged power of a [[dipole antenna]] has been replaced by its full form.

===== Impedance of free space =====
The ** radiation resistance of free space** can be found by using the definition of [[Poynting flux]] in [[Gaussian units]]:

$$ S = \frac{c}{4\pi} E^2 = \frac{cV^2}{4\pi l^2} = \frac{V^2}{R_0 l^2} $$

because $E=V/l$ and the **impedance of free space**

$$ R_0 = \frac{4\pi}{c} = \frac{1}{c\epsilon_0} = 377 \ \Omega $$

because $\epsilon_0=1/(4\pi)$ has to be used while converting from Gaussian to SI units. The definition of impedance of free space has to be this because $P=V^2/R$ which entails $S=V^2/(Rl^2)$ because flux is power per unit area.

A black hole has the same radiation resistance as free space because it is a perfect absorber. So a black hole totating in an external magnetic field can generate electricity with $V/I$ ratio of 377 $\Omega$ which might power the jets of a [[quasar]].