The inverse square law is a principle in physics that describes how the strength of a field, radiation, or wave spreads out from the source. If it sounds like a complicated idea, don’t worry. The inverse square law is actually more intuitive than you may think. Here is an example.
Suppose you wake up one morning and can’t find your phone. After a few unsuccessful attempts, you ask a friend to call you on their phone so you can track your phone with the ringing sound. You find it almost instantly – under the pillow.
In this case, the inverse square law helped you zero in on your phone. With every step you took toward the pillow, the phone got twice as loud. It’s like nature was working for you!
In fact, this concept is so intuitive that we can precisely estimate how loudly we should speak to catch someone else’s attention with our voice. If someone is very close, whisper. If someone is close enough, speak. If they are down the hall, yell. If they are two blocks down the street, get closer first.
We are always balancing the variables of distance and intensity. Which is the essence of the inverse square law.
While many phenomena obey the inverse square law, this article discusses how this principle applies specifically to gravitation, a force we understand intuitively.
What is the inverse square law?
The inverse square law is a principle that describes how the strength of a physical quantity spreads out in all directions from the point source.
According to the inverse square law, “the intensity of a given physical quantity is inversely proportional to the square of the distance from the source of that physical quantity.”
In mathematical terms,
$$Intensity ∝ 1/〖Distance〗^2$$
Although we’ve used sound energy to explain the inverse square law, this principle applies across various physical quantities. This includes:
- Gravitation – the strength of gravitation decreases with the square of the distance from the body.
- Electricity – the strength of attraction or repulsion between two electrically charged particles decreases with the square of the distance between them – this is Coulomb’s law.
- Electromagnetic radiation – the intensity of electromagnetic radiation (for example, light) decreases with the square of the distance from the point source
- Sound energy – loudness increases four times when you half the distance between the source and receptor.
This article focuses on how the inverse square law applies to gravitation taking inspiration from Sir Isaac Newton himself. And for that, let’s revisit one of the most famous tales in the history of physics – Isaac Newton and the falling apple story.
One Giant Intellectual Leap: From the Apple to the Moon
The inverse square law of gravitation was discovered by Sir Isaac Newton when he extended the force of gravity from Earth to the Moon. Here’s how it happened.
Apparently, Newton was sitting under an apple tree one day and began pondering why apples fall down. Clearly, the force of gravity is pulling them but how far up does this force extend?
You see, unlike other forces, gravity doesn’t require physical contact to act on an object. This means it can pull an apple from a tree, be felt on the tallest of mountains, and even draw condensed water droplets from the atmosphere, causing rain—all without direct contact.
It seemed as if gravity didn’t have a roof over its head, which led Newton to come up with a brilliant supposition. What if it goes as far as the Moon?
If the force of gravity reaches the Moon, will it be as strong as it is here on Earth, or will it weaken with distance? And if it weakens, how does its strength vary with distance from Earth?
The answer to these questions is the inverse square law.
Newton’s Cannonball
According to Isaac Newton, if the force of gravity reaches all the way to the Moon, then like the apple and the rain, it should also fall toward Earth. But it doesn’t. Instead, it revolves around it – why?
To explain this question, Newton developed a brilliant thought experiment to explain the motion of the Moon around the Earth.
He envisioned a cannonball being fired horizontally from a very tall mountain. In his experiment, he initially fired the cannonball at a regular speed, and it traced a parabolic path, eventually falling back to Earth (Path A).
He then bumped up the speed of the cannonball each time he fired and assumed there was no air resistance so that gravity was the only force acting on the cannonball. In his experiment, it reached a point where the rate of fall of the cannonball coincided with the curvature of the Earth underneath it.
Thus, locking the cannonball in a circular motion around the Earth – ever falling but not making contact.
Newton theorized that this is how the Moon moved around the Earth.
Deriving the inverse square law – Newton’s way
On Earth, objects accelerate downwards at an acceleration of 9.8m/s2. If the force of gravity extends to the Moon, will it exert the same strength as it does on Earth, or will it be weaker?
We can answer this question by comparing the acceleration of the Moon to the acceleration of (say) an apple on Earth.
This is how Isaac Newton approached the problem: Using astronomical data available at the time, Newton determined the Moon’s distance from Earth (the radius of its orbit) and calculated the centripetal acceleration required to keep the Moon in its orbit.
Upon comparison, he noticed that the acceleration of the Moon was 1/3600 of the free fall acceleration of an object at the Earth’s surface. The Moon’s orbit radius was about 60 times the radius of the Earth. And 3600 is 60 squared. This led him to the inverse square law.
Deriving the Inverse Square Law – the modern way
We can revisit Newton’s arguments using present-day data as follows:
First, we assume that acceleration due to gravity varies with distance from Earth. The further you move away from Earth; the less powerful gravity becomes.
We can express this relationship in crude mathematical terms as follows:
Acceleration due to gravity varies proportionally to some power of the distance from the Earth. That is,
$$a α r^n$$
Which becomes,
$$a=kr^n………. Equation (1)$$
We keep that equation in the parking lot for now and shift our attention to the Moon.
So the Moon revolves around the Earth from an average distance of 384,400km, completing one revolution in 28 days. We can calculate the centripetal acceleration as follows,
$$a= v^2/r$$
where,
$$v= 2πr/T$$
Combining the two,
$$a= 4πr/T^2 = (4 x 3.142 x 384,400km)/〖28 days〗^2$$
$$a=0.00259 m⁄s^2 …….. Equation (2)$$
Comparing this with the acceleration due to gravity on the surface of the Earth (9.8m/s2), we get,
$$a_moon/a_earth = (0.00259 m⁄s^2 )/(9.8 m⁄s^2 )=0.000264 ≅ 1/3600$$
thus,
$$a_moon/a_earth =1/3600………Equation (3)$$
At this point, we revisit ‘Equation 1’ and calculate the ratio amoon/aearth
$$a_moon/a_earth = (kr_m^n)/(kr_e^n )= (r_m^n)/(r_e^n )= (r_m/r_e )^n$$
Equating the right-hand-side of the equation amoon/aearth with 1/3600, it becomes,
$$(r_m/r_e )^n= 1/3600$$
$$Then substituting the known values for rm = 384,400km and re = 6370km, we get
$$(384,400km/6370km)^n= 1/3600$$
$$(60.3)^n= 60^(-2)$$
Thus, rounding off 60.3 to 60, we get
$$n= -2$$
This would imply the acceleration due to gravity varies with distance from the center of the Earth according to ‘Equation 1’ or
$$a= k/r^2$$
which becomes,
$$a α 1/r^2$$
Where r is the distance from the center of the Earth.
Points to note:
- You may be wondering whether the exponent “2” on “r” is exactly 2 and not an approximation. Like, say, 2.001 or 1.998. The truth is, we don’t know. All that we know is that the law has been experimentally consistent so far (save for a few scenarios) and there is no reason to think that “r” is raised to anything other than “exactly 2”.
- To reach this conclusion, Newton assumed the mass of the Earth to be concentrated at its geometric center. This can actually be rigorously proved if we assume the Earth to be spherical (a fair assumption) and that the distribution of its mass may change with distance from the center but not with angular coordinates (also a good assumption).
Conclusion
The inverse square law is one of the most important laws in physics. Not just in gravity but in many other fields.
Whilst, there are many ways to derive it, I’ve focused on Isaac Newton’s approach as it relates directly to gravity (and of course, it is Isaac Newton!). As always, constructive criticism is welcome.
