d~h, For the small the stable state did not occur. Film continued to stretch, discharge developed and film exploded. Figure 3. Let us explain this theoretically. In such interaction the electrostatic is balanced by the surface tension force. The latter can be calculated by means of Laplace equation while the electrostatic force between the charge and the film can be calculated using an "electrostatic image method". Equating the Coulomb and the h surface tension forces we can calculate the dependence of on the sphere charge Q d. and We will use the approximation, when the deformation is much less than the loop radius and the distance to the charged sphere. Electric force. The external redistributes the charges on the film; The film surface is equal-potential; The interaction between the charge and the grounded sphere film can be "effectively" presented as the interaction between the two point charges; The value and the place of "effective" charge must give equipotential film surface; We assume that the film is a sphere h segment ABC of the height a. AC the loop (frame) of a radius – Q the external charge, at the distance – d>>h q’ the "effective" charge at the – x distance from the loop. = = 0 From the Figure 4 the condition of φB φC (a grounded film) leads to the following values: ; , (1) and the effective electrostatic interaction force will be: . (2) This force must be balanced by the surface tension force. Surface tension force. Pl = 4σ0 / R Using the Laplace formula and taking into account, that we obtain . If we write down Fl = pS we will get that the force of the surface tension is , . (3) Balance of Electrostatic and Surface tension forces. Fk=Fl the force equilibrium condition for (2) and (3) we get: . (4)
For the small d~h, the stable state did not occur. Film continued to stretch, discharge developed and film exploded. Figure 3. Let us explain this theoretically. In such interaction the electrostatic is balanced by the surface tension force. The latter can be calculated by means of Laplace equation while the electrostatic force between the charge and the film can be calculated using an "electrostatic image method". Equating the Coulomb and the surface tension forces we can calculate the dependence of h on the sphere charge Q and d. We will use the approximation, when the deformation is much less than the loop radius and the distance to the charged sphere. Electric force. The external redistributes the charges on the film; The film surface is equal-potential; The interaction between the charge and the grounded sphere film can be "effectively" presented as the interaction between the two point charges; The value and the place of "effective" charge must give equipotential film surface; We assume that the film is a sphere segment ABC of the height h AC – the loop (frame) of a radius a. Q – the external charge, at the distance d>>h q’ – the "effective" charge at the distance x from the loop. From the Figure 4 the condition of φB = φC = 0 (a grounded film) leads to the following values: ; , (1) and the effective electrostatic interaction force will be: . (2) This force must be balanced by the surface tension force. Surface tension force. Using the Laplace formula Pl = 4σ0 / R and taking into account, that we obtain . If we write down Fl = pS , we will get that the force of the surface tension is . (3) Balance of Electrostatic and Surface tension forces. the force equilibrium condition Fk=Fl for (2) and (3) we get: . (4)
2010_4_Soap_film_Georgia_AB_v1.docx