### Fall 2007

Fall 2007 1 Mech

Fall 2007 2 Mech

Fall 2007 3 EM

Fall 2007 4 EM

Fall 2007 5 QM

Fall 2007 6 QM

Fall 2007 7 SM

Fall 2007 8 SM

Fall 2007 9 Misc

Fall 2007 10 Misc

Fall 2007 11 Mech

Fall 2007 12 Mech

Fall 2007 13 EM

Fall 2007 14 EM

Fall 2007 15 QM

Fall 2007 16 QM

Fall 2007 17 SM

Fall 2007 18 SM

Fall 2007 19 Misc

Fall 2007 20 Misc

In the absence of other forces, surface tension causes a liquid droplet to assume a spherical shape. Lord Rayleigh has shown that this is no longer true for an electrified droplet of a sufficiently large charge *Q*. (This instability has found a practical application in ink-jet printers.) Compute the corresponding critical charge *Q _{c}* for a droplet of radius

*R*and surface tension $\sigma$.

Hint: The capacitance C of a nearly spherical object is related to its surface area S by the Aichi-Russel formula

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Answer

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The total energy of the droplet is

(2)Function *E(S)* has the minimum at *S = Sc*,

However, since the sphere has the mimimal surface area for a given fixed volume, *S* cannot be smaller than $4 \pi R^2$. As a result, for small *Q* the sphere remains the optimal shape. The critical charge is determined by the condition $S_c = 4 \pi R^2$, which gives:

in agreement with Lord Rayleigh (1882). At *Q* somewhat larger than *Q _{c}* the droplet deforms into a prolate ellipsoid. This is the answer the student is expected to give for this problem.

Actually, an astute reader may realize that this answer may be incomplete. In principle, the droplet can also change its shape discontinuosly, e.g., by splitting into two smaller droplets. Let us examine this "first-order transition" scenario assuming the new droplets are also spherical and equal in size.

For the droplet of charge *q* and radius *r* the energy is

Comparing the energies of one droplet with *q = Q* and *r = R* with that of two droplets with *q = Q/2* and *r = R/2 ^{1/3}*, we conclude that the first-order instability occurs at

We see that *Q _{m} < Q_{c}*, and so the first order transiton wins. More precisely, the spherical droplet with charge

*Q*is metastable, and so in practice it still may have a long lifetime.

_{m}< Q < Q_{c}
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