Question

Suppose that a force or energy originates from a point source and spreads its influence equally in all directions, such as the light from a lightbulb or the gravitational force of a planet. So at a distance r from the source, the intensity I of the force or energy is equal to the source strength S divided by the surface area of a sphere of radius r. Show that I satisfies the inverse square law \(I = \frac{k}{{{r^2}}}\), where \(k\) is a positive constant.

Short answer

It is proved that \(I\) satisfies the inverse square law \(I = \frac{k}{{{{\mathop{\rm r}\nolimits} ^2}}}\).

Step-by-step solution

Inverse square law

The quantity is modeled by a function of the form \(f\left( x \right) = \frac{C}{{{x^2}}}\). It is calledinverse square law.

Show that I satisfies the inverse square law 

a)

The surface area of a sphere is given by\(S = 4\pi {{\mathop{\rm r}\nolimits} ^2}\).

The intensity of the force is shown below:

\(\begin{aligned}I &= \frac{S}{{4\pi {{\mathop{\rm r}\nolimits} ^2}}}\\ &= \left( {\frac{S}{{4\pi }}} \right)\left( {\frac{1}{{{{\mathop{\rm r}\nolimits} ^2}}}} \right)\\ &= \frac{{\frac{S}{{4\pi }}}}{{{{\mathop{\rm r}\nolimits} ^2}}}\end{aligned}\)

Therefore, the intensity of the force\(I = \frac{k}{{{{\mathop{\rm r}\nolimits} ^2}}}\) with a positive constant\(k = \frac{S}{{4\pi }}\).

Thus, it is proved that \(I\) satisfies the inverse square law \(I = \frac{k}{{{{\mathop{\rm r}\nolimits} ^2}}}\).