How to show field strength varies as 1/r
If you have an isotropic antenna it pushes power out equally in all directions and this means, at an arbitrary distance of "R" all the power is travelling thru a sphere of radius "R". A receiving antenna is said to have an "effective area" (or aperture) which picks-up a fraction of the transmitted power. At twice "R" the surface area of the sphere is 4x as big hence power received into the same antenna is quartered. Surface area of a sphere is \$4\pi R^2\$

Hence power reduces with distance squared. The two fields associated with EM power are the electric and magnetic fields. These combine to yield "power" according to their mathematical product and, if one halves, then so does the other. There is no compromise on this - free-space has a radiation resistance (just like coax has a characteristic impedance that is resistive) and E and M waves are proportioned to each other with a resistance of 377 ohms (see this wiki article).

Thus, a halving of E field produces a halving of the H (magnetic) field and therefore a quartering of the power. A quartering of power is when distance doubles so therefore when distance doubles the E field (or the H field) halve in amplitude.

All you've got to do is convince yourself that my explanation of power quartering with a doubling of distance is bona fide! Here's a picture of a real antenna radiation pattern that might help: -



The above picture is for the classic dipole antenna - the third diagram (c) shows (like a sort of doughnut with very tiny hole), equal-power levels distributed at a certain distance. As distance increases, the shape remains the same but the power per square metre at the surface reduces as the square of the distance because of the physical area of the doughnut.

For a dish antenna, the same \$\frac{1}{d^2}\$ rule applies with one exception. If the receiving dish is bigger than the transmitting dish it can completely "capture" all the transmitted power if the distance isn't too great - power received will remain theoretically constant up to this point then, as the transmiter's lobe area starts to fall outside the effective aperture of the receiving dish, power will reduce as per distance squared.