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Why is the strength of a field (i.e. gravitational or electric) inversely correlated to the square of the radius?

It is clear to me why the further you get a way from a field, the field's enacted force decreases. However, this makes me hypothesize that the relationship should be:

$$F ~=~ k/r$$

However, it is in fact:

$$F ~=~ k/r^2$$

Is this related to the fact that the surface area of a sphere is a function of the square of its radius?

An intuitive explanation would be very much appreciated.

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  • $\begingroup$ Because Newton said so. $\endgroup$
    – evil999man
    Apr 9, 2014 at 12:09
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    $\begingroup$ Your guess that it's related to the fact that the surface area of a sphere is a function of the square of its radius is correct. Not all fields drop off as $1/r^2$, but all fields that conserve flux and which come from point sources will. $\endgroup$ Apr 9, 2014 at 12:17
  • $\begingroup$ Related: physics.stackexchange.com/q/22010/10851 , physics.stackexchange.com/q/48447/2451 and links therein. $\endgroup$
    – Qmechanic
    Apr 9, 2014 at 16:48

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