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