Tag: For laymen

Let us recall Snell’s well-known law, which describes the phenomenon of refraction:
– a ray of light undergoes a deviation when it passes from one transparent medium to another with a different refractive index (see Figure 2).

Figure 1 – Snell law

Figure 2 – Refraction example

Mathematically (see Figure 1):
[1]    n’ sin(θ’) = n” sin(θ”)
where  n’ ,  n”  are the refractive indices of the two transparent media,  θ’ ,  θ”  are the angles of incidence and refraction (measured relative to normal at the interface between the media).
In other words, given two media, the ratio between the respective sines of the angles of incidence and refraction is constant.
A first proof can be obtained by applying Pierre de Fermat’s (1601-1665) principle of least time, namely that light follows the path that requires the least time to travel from one point to another. Attached below is the simple geometric-analytical proof (in italian):

  Fermat_Snell_geometric-analytical demonstration

Okay, this relationship (perhaps already studied in the 10th century by the Persian mathematician Ibn Sahl) is certainly interesting, but what is the physical motivation for this constant quantity? It is necessary to identify where this regularity lies.
The reasons derive directly from Maxwell’s equations, the fundamental laws describing the interaction between electric and magnetic fields.
From these, it follows that the boundary conditions require that the tangential components of the electric field and magnetic field be continuous across the separation surface. Therefore, the phase of the incident, (reflected,) and transmitted wave must be the same along this boundary.

The assumption that the frequency ( ν ) of the incident light remains constant (i.e., that it is independent of the medium in which it propagates) can also be understood by considering that if this were not the case, it would lose coherence as it crossed the surface between the two media, probably transforming the image observed through the separation surface into a uniform grey color, which is not observed.
Furthermore, since the energy associated with incident photons is  E = h ν , where  h  is the Planck constant, the same conservation of energy tells us that the frequency cannot change.
Quantistic note: this consideration refers to a single photon, which would change color (towards red) if it lost energy (phenomenon that does not occur for all observed refracted photons). Considering a beam of photons, however, it may be that the second medium causes absorption of the incident light, thus a decrease in total energy, but this concerns the entire beam that absorbs some photons completely and not the individual photons.

Considering the wave vectors  k’  (incident) and  k”  (refracted) associated with their respective phases, and equating their projections on the separation interface, we obtain:
[2]     k’ sin(θ’) = k” sin(θ”)
but  k = ω / v  , where  ω  is the angular frequency of the electromagnetic wave of the incident light,  v  is the velocity in the medium, and defining the refractive index  n = c / v  (derived from the constants  ε₀  and  μ₀  of the medium using Maxwell’s equations, where  c =  1/√(ε₀ μ₀)  ), from  [2]  we finally obtain  [1] .

Archimedes of Syracuse (287-212) was perhaps one of humanity’s greatest mathematical geniuses. He is remembered for the principle of hydrostatic thrust, the screw pump, and war machines, but his most advanced studies were on conics and mathematical methods for calculating surfaces and volumes (e.g., of sphere). These are considered among the highest achievements of ancient science and, although they lack modern formal rigor, they represent the direct precursors of the concept of limits and integral calculus (independently developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century). See, for example, Lucio Russo, Archimedes: A Great Ancient Scientist (Carocci Editore, 2019).

One day, my young son asked me, “Why do ships that are heavy and made of iron float?” I couldn’t explain the density of water to him, but I found a good answer for a child: “Because they weigh less than the water they displace.”
This is exactly Archimedes’ principle of hydrostatic thrust, which in more complicated terms is expressed as: “A body immersed in a fluid experiences an upward thrust equal to the weight of the volume of fluid displaced.” So if this is greater than the weight of the body, it floats.

We can express Archimedes’ principle of hydrostatic thrust (force) F(A) using the following formula:
[1]    F(A) = m * g = ρ(fluid) * V(displaced) * g
where  m =  ρ *  V  is the mass,  g  is the acceleration due to gravity,  ρ  represents the density, and  V  represents the volume of fluid displaced by the body.
For this reason, any body also ‘weighs’ according to the density of the fluid in which it is immersed. The greater the density, the greater the hydrostatic thrust and the lower the weight of the body.

As an example, if an iron nail (which, due to its shape, does not seem to be able to float like a ship by displacing a lot of fluid) were placed in a container with liquid mercury, which has a density almost double that of iron, the nail would float at the level of the weight of mercury displaced. Calculations show that it would be immersed just over half its volume (about 58%).

In this video, we can see how an iron anvil floats in mercury because the density of iron (7.87 g/cm³) is almost half that of mercury (13.55 g/cm³) (source: Wonder of science on X channel).

Another famous example of buoyancy is that of hot air balloons: they float exactly like a ship because the volume of fluid (atmospheric air) they displace weighs more (is denser) than that contained in the balloon, which is heated with a burner (and is therefore less dense).

An hot air balloon (source: freedome.it)