An idea that was always hard for me to grasp intuitively was buoyancy. I’d heard the common explanations but I could never understand what the relavant characteristics were. It started with…

Act 1: Archimedes

As the age-old maxim goes “the magnitude of the buoyant force is equal to the weight of the water displaced”. Here’s a fun story about Archimedes you may not have heard. I’ll replay the argument here:

Imagine a submerged ball and the water around it. There is pressure due to the external water pushing the ball up. Now suppose we replace our ball with a water ball of the same shape. Since the water ball is exactly supported by the difference in pressure (no net force on the water ball) and the solid object experiences the same pressure environment, it follows the buoyant force on the solid object is equal to the weight of the water ball!

This, of course, gives:

\[\vec{F}_{b} = m_{water}g\hat{z}\]

But for a curious mind, this leaves a couple of non-obvious questions. Why is the shape of the submerged ball not relavant? Does it matter if an ball is partially submerged? What forces would an insect sitting on the surface of our ball experience? Does it matter where the insect sits?

We look forward…

Act 2: Gauss

Another way to understand buoyancy, perhaps unexpectedly, is through Gauss’ Law in electrostatics. Gauss tells us the electric flux through a closed surface \(S\) is proportional to the enclosed charge \(q_{enc}\).

Picture a beach ball submerged in a pool of water and the forces it experiences on its surface. As a shift of perspective, consider an analgous uniformly charged sphere in an electric field (instead of a pressure field) pointing upward. What does this field look like?

We know for a fluid, by taking the force per unit area:

\[\vec{P} = -\rho_{water}g\vec{z}\]

where \(\vec{z}\) is a vector pointing upward with magnitude equal to its depth below the surface. Applying the Divergence Theorem,

\[F_{b} = -\iint\limits_{S} \vec{P} \cdot \vec{da} = -\iiint\limits_{V} (\nabla \cdot \vec{P}) dV = \rho_{water}g\iiint\limits_{V} dV\]

We see this is an identical result to above. The intuition is identical to that of the Divergence Theorem or Gauss’ Law: any pressure gradient (think charge) that would be created inside the sumberged object is equal the pressure flux (ie. force) on the ball. Adding these differential flux vectors on the surface of the ball gives us the net force on the ball.

We now see that the shape of the ball is irrelavant (differently shaped objects may have different surface force elements \(\vec{P} \cdot \vec{da}\) but cancellation ensures they’ll add to the same flux) and the differential force experienced by an insect on the ball is just \(\rho_{water}g\vec{z} \cdot \vec{da}\) (which varies across the surface of the ball). Partial submersion is not relevant except that we may consider atmospheric pressure (in the same way!) on the revealed portion of the object (negligible, \(\rho_{atm} \ll \rho_{water}\)).

Postlude: What makes a scientific explanation?

One may reflect: Where do these explanations differ? What makes a good explanation?

We should take a moment to appreciate the ingenuity of Archimedes’ argument, developed almost 2 millennia before the invention of calculus. At the same time, Gauss’ Law gives a more mechanistic explanation, providing intuition and allowing us to answer several related questions.

We see that ideas can be clever and correct, but not explanatory. Archimedes’ argument feels a lot like a proof by induction to me. Although correct, it doesn’t give me a feeling of why the result is true morally.

Explanations in science are just descriptions at the end of the day. It’s important when communicating our ideas to impart a type of moral description of why something ought to be true, leaving the listener to only clear the weeds on their own.