Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 355 / 2 / a

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- Solution a

Solution

For a cylinder of radius

r

L

, the mean curvature is

H = 1/ (2 r)

and the area is

2 π r L

. Its Helfrich membrane energy per unit length is therefore

\frac{E _{m}}{L} = 2 π r (σ + \frac{k _{c}}{2 r ^{2}}) = 2 πσ r + \frac{π k _{c}}{r} .

The first term favors a narrow tube and the second penalizes its curvature. Setting the derivative with respect to

r

to zero gives

2 πσ - \frac{π k _{c}}{r ^{2}} = 0, r_{0} = \frac{k _{c}}{2 σ} .

The positive second derivative confirms a minimum.

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