Young–Laplace equation

Remarks

The idea of the proof: surface tension is the energy per unit area of the interface. With the sign convention above, the first variation of area under a displacement of $S$ by $\delta n$ is $\int_S 2H\delta \, dA$. At equilibrium the work $\int_S (p_- - p_+)\delta\,dA$ done by the pressure difference must balance the change $\sigma\,\delta(\text{area})$ for every $\delta$, which gives the equation pointwise. Example: for a spherical drop of radius $R$ with $n$ pointing outward, $\kappa_1 = \kappa_2 = 1/R$, so the pressure inside exceeds the pressure outside by $2\sigma/R$. A soap bubble has two surfaces, so its excess pressure is $4\sigma/R$. A soap film with equal pressures on both sides has $H = 0$, so it is a minimal surface.

Going deeper

For a student

The equation is the boundary condition for the normal stress at a fluid interface at rest, and it enters the dynamic condition of capillary waves. Combined with the hydrostatic pressure it gives the shape of menisci, pendant drops and the capillary rise in a thin tube. When the pressure difference is constant, it says that interfaces in equilibrium are surfaces of constant mean curvature. This links fluid statics to the differential geometry of such surfaces.