Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== A thin spherical membrane with isotropic surface tension TTT (force per unit length, N/m) produces an inward pressure Δp\Delta pΔp related by the Laplace formula for a sphere: ===

Δp  =  2TR.\Delta p \;=\; \frac{2T}{R}.Δp=R2T.
(Here Δp\Delta pΔp is the pressure difference across the membrane; to hold the shell you need inward Laplace pressure equal to electrostatic outward pressure.)

Set equilibrium: 2T/R=pe2T/R = p_{\rm e}2T/R=pe. Solve for TTT:

  T  =  pe R2  =  Q264π2ε0 R3  .\boxed{ \; T \;=\; \frac{p_{\rm e}\,R}{2}
\;=\; \frac{Q^{2}}{64\pi^{2}\varepsilon_0\,R^{3}} \; }.T=2peR=64π2ε0R3Q2.
This is the tension required to exactly balance the Coulomb self-repulsion.