# Soap Film between Two Equal and Parallel Rings

Soap Film between Two Equal and Parallel Rings

A soap film is formed between two parallel rings of radius separated by a distance . To minimize the surface-tension energy of the soap film, its total area seeks a minimum value. The derivation of the shape of the film involves a problem in the calculus of variations. Let represent the functional form of the film in cylindrical coordinates. The area is then given by . The integrand is determined by the Euler–Lagrange equation -=0, which can be reduced to its first integral , a constant. The solution works out to , a catenary of revolution, with the boundary condition . When , the film collapses to disks within the two rings.

r

d

S

ρ(z)

S=2πρ(z)

d/2

∫

-d/2

1+

dz2

(ρ'(z))

F(ρ,ρ')

z

∂F

∂ρ'

∂F

∂ρ

ρ'-F=c

∂F

∂ρ'

ρ(z)=ccosh(z/c)

r=ccosh(d/2c)

S<2π

2

r