Exercise 1.2.36

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 $I = k/r^2$, where $k$ is a positive constant.

Khagan Gulusoy · Accepted Answer

Since the energy from a point source spreads its influence equally in all directions, we have:
$$I=\frac{S}{S_{	ext{surface}}}.$$
$$I=\frac{S}{4\pi r^2}.$$
$$I=\frac{S}{4\pi}\cdot\frac{1}{r^2}.$$
If we denote the fixed quantity $\frac{S}{4\pi}$ by the constant $k\in \mathbb{R}_+$, we get:
$$I = k/r^2.$$

Exercise 1.2.36

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Exercise 1.2.36

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