Soap Film between Two Equal and Parallel Rings

separation d

8

radius r

6

A soap film is formed between two parallel rings of radius

r

separated by a distance

d

. To minimize the surface-tension energy of the soap film, its total area

S

seeks a minimum value. The derivation of the shape of the film involves a problem in the calculus of variations. Let

ρ(z)

represent the functional form of the film in cylindrical coordinates. The area is then given by

S=2π

d/2

∫

-d/2

ρ(z)

1+

2

(ρ'(z))

dz

. The integrand

F(ρ,ρ')

is determined by the Euler–Lagrange equation



z

∂F

∂ρ'

-

∂F

∂ρ

=0

, which can be reduced to its first integral

ρ'

∂F

∂ρ'

-F=c

, a constant. The solution works out to

ρ(z)=ccosh(z/c)

, a catenary of revolution, with the boundary condition

r=ccosh(d/2c)

. When

S<2π

2

r

, the film collapses to disks within the two rings.