Editing Free-space path loss (section)

==Derivation==
The radio waves from the transmitting antenna spread out in a spherical wavefront. The amount of power passing through any sphere centered on the transmitting antenna is equal.  The surface area of a sphere of radius <math>d</math> is <math>4\pi d^2</math>.  Thus the intensity or power density of the radiation in any particular direction from the antenna is inversely proportional to the square of distance   
:<math>I \propto {P_t \over 4\pi d^2}</math>
(The term <math>4\pi d^2</math> means the surface of a sphere, which has a radius <math>d</math>. Please remember, that <math>d</math> here has a meaning of 'distance' between the two antennas, and does not mean the diameter of the sphere (as notation usually used in mathematics).)  
For an [[isotropic antenna]] which radiates equal power in all directions, the power density is evenly distributed over the surface of a sphere centered on the antenna
:<math>I = {P_t \over 4\pi d^2} \qquad \qquad \qquad \text{(1)}</math>
The amount of power the receiving antenna receives from this radiation field is 
:<math>P_r = A_\text{eff}I \qquad \qquad \qquad \text{(2)}</math>
The factor <math>A_\text{eff}</math>, called the ''effective area'' or ''aperture'' of the receiving antenna, which has the units of area, can be thought of as the amount of area perpendicular to the direction of the radio waves from which the receiving antenna captures energy. Since the linear dimensions of an antenna scale with the wavelength <math>\lambda</math>, the cross sectional area of an antenna and thus the aperture scales with the square of wavelength <math>\lambda^2</math>.<ref name="Cerwin" />  The effective area of an isotropic antenna (for a derivation of this see [[antenna aperture]] article) is 
:<math>A_\text{eff} = {\lambda^2 \over  4\pi}</math>
Combining the above (1) and (2), for isotropic antennas
:<math>P_r = \Big({P_t \over 4\pi d^2}\Big)\Big({\lambda^2 \over 4\pi}\Big)</math> 
:<math>\text{FSPL} = {P_t \over P_r} = \Big({4\pi d \over  \lambda}\Big)^2</math>