11.6. Inverse Square “laws”

Now you’ve seen a very familiar behavior of two models in nature, Newton’s Gravitational law and Coulomb’s law. Both have the same abstract form:

\[ F_{12}=\text{Constant}\frac{\text{Source}_1\text{Source}_2}{R_{12}} \]

Here we’re to imagine two “sources,” named One and Two, which are separated by a distance \(R_{12}\) that experience a force, \(F_{12}\) between them. Obviously:

• Gravitational force: Source\(_i\) = \(M_i\), a mass, in units of kilograms

• Electrical force: Source\(_i\) = \(Q_i\), an electrical charge, in units of Coulombs

Some characteristic fundamental Constant sets the scale for how large such a force might be. If Newtons are our force units, then:

• Gravitational force: “Constant” = \(G=6.673×10^{-11}~\text{ Nm}^2/\text{kg}^2\)

• Electrical force: “Constant” = \(k_E=9 \times 10^9 \text{ Nm}^2/\text{C}^2 \nonumber\)

Perhaps you’re not surprised that there’s also an “inverse-squared” rule for magnetic forces? We don’t talk about it very often because it’s the force between two magnetic poles, which we don’t ever see isolated in nature.

But Mr Coulomb also studied that force with his sensitive balance by suspending a bar magnet from its center and then using the pole of a long second magnet to attract or repel the suspended one. So we then have:

• Magnetic source: Source\(_i\) = \(m_i\), a unit magnetic pole, in units of Ampere-meter

• Magnetic force: “Constant” = \(k_m=1 \times 10^{-7} \text{ N}/\text{A}^2 \nonumber\)

Okay. That’s unfortunate. Yet another use of the 19th letter off the alphabet, m. And, a current has crept into this discussion of magnets as we just learned, “Amperes” is a measure of electrical current (“amps”). So this is indeed a cumbersome item.

11.6.1. Thing One

Let me let the cat out of the bag (although in my opinion, bags are a good place to have cats…I know. Sorry.) by looking at these two constants, \(k_E\) and \(k_m\). First let me replace the unit of Ampere with its more basic unit of Coulomb/second.

\[ k_m=1 \times 10^{-7} \text{ N}/\text{A}^{-2} = 1 \times 10^{-7} \text{ N}\text{C}^{-2}\text{s}^2 \nonumber \]

Let’s relate them now as a ratio:

\[\begin{split} \begin{align} \frac{k_E}{k_m} &= \frac{9 \times 10^9 \text{ Nm}^2/\text{C}^2}{1 \times 10^{-7} \text{ N}\text{C}^{-2}\text{s}^2} \\ &= 9\times 10^{16} \;\;\text{m}^2/\text{s}^2 \end{align} \end{split}\]

Can you see that this last line is the square of a velocity? If we take the square root of this:

\[ \sqrt{\frac{k_E}{k_m}} = 3 \times 10^8\;\;\text{m}/\text{s} \]

A particularly famous speed…that of light. So stay tuned for this as it will be a big deal in Lesson 14.

In fact, this will be the biggest deal of the 19th century and be the direct inspiration to Einstein and his Theory of Relativity.

11.6.2. Thing Two

Why is this inverse square such a thing?

Let’ suppose that you’re going to paint the inside of a beach ball by exploding a very precisely constructed ball of paint.

Wait. Why would you do that?

Glad you asked. It’s a metaphor to make a complicated mathematical calculation a simple mind-game.

You detonate your paint-bomb and droplets of paint fly out in all directions uniformly coating the inside. We could estimate the darkness of the paint covering by counting the number of little drops that uniformly stuck to the inside.

Wait. Again?

Glad you asked. Stop it.

Suppose that worked well and we want to do it to a beach ball that’s twice the size of the original, but using the same paint-bomb. Would the darkness be the same? Darker, or lighter?

Of course, it’s going to be lighter by how much? By a factor of 4 less. That’s because the area of a sphere (the area of the inside of the beach ball) is $\( A(\text{sphere}) = 4 \pi R^2 \)\( If you increase \)R$ by a factor of 2 then the area becomes a factor of 4 bigger. But if the paint drops uniformly coat the bigger sphere, then there are going to be 4 times fewer of them per unit area than for the half-sized beach ball.

That’s the essence of the “inverse square-law” for any of these three different models of physics. It implies something about an emanation (the “paint”) and it’s a statement totally about geometry, the fact that the area of a sphere is what it is for three space dimensions.

Suppose space had more than 3 dimensions? Up, down, in and out and…over there? Then that “2” in the denominator of each of those relationships would be something other than two. And boom. We’ve got a whole field of experimental research: trying to measure that exponent very well to check that we have 3 dimensions in space or something different.