return 9. An eiectron bubble in liquid helium (adapted from APhO 2010 Taipei) Wh

Question

return 9. An eiectron bubble in liquid helium (adapted from APhO 2010 Taipei) When an electron is planted inside liquid helium, it can repel the atoms around it isotropically, thus forming a spherical surface, inside which contains nothing but the electron itself. This is known as an electron bubble. The components of an electron's position vector 9 = (x,y, z) and momentum vectorji = (Pa, pvp.) must obey Heisenberg's uncertainty relations 9Ap h/4n, where h is the Planck constant. Note that the uncertainty of a variable f, denoted by /is given by where, the overhead bar denotes the average (mean) of the quantity under it. ider an electron bubble in liquid helium with an equilibrium radius R. The electron of mass m moves freely inside the bubble with kinetic energy Ex and exerts a pressure Pe on the inner side of the bubble-liquid interface. The pressure exerted by iquid helium on the outer side of the interface is PHe: The liquid is kept at a constant temperature close to 0 K and the surface tension is given by 3.75 × 10-4Nm_, . (a) Find a relation between Be, Fe and . [2 marks (b) Find a relation between Ex and Pe. 4 marks) (c) Let Eo be the smallest possible value of EK consistent with Heisenberg's uncertainty relations when the electron is inside the bubble of radius R. Estimate Bo as a function of R. [4 marks] (a) Let Pa be the equilibrium radius of the bubble when Ex o and Phe0. Obtain an expression for R,and 2 marks calculate its value.

Explanation / Answer

a. given, electron bubble in liquid helium

pressure outside the bubble = Phe

Pressure inside the bubble = Pe

surface tension, sigma = 3.75*10^-4 N/m

Equilibrium radius of bubble = R

Kinetic energy = Ke

mass of electorn = m

so for a bubble with surface tension sigma

Pe - Phe = 4*sigma/R

b. for an electron with kinetic energy Ek

pressure inside the bubble created by electorn = Pe

then Pe = Ek/Volume ( volume of the bubble)

so Pe = 3Ek/4*pi*R^3

c. if Eo is the smallest value of Ek then

from heisenberg's uncertianity principle

dx*dp > h/4*pi

where dp is error in momentum

dx is error in position

now, p^2 = 2Em

p = sqroot(2Em)

then dp/p = dE/2E

hence, dx*dp > h/4pi

becomes

dx*dE*sqroot(2Em)/2E > h/4*pi

dx*dE*sqroot(m/2E) > h/4*pi

now, dE = Eo ( given )

then dx = 2R

so, 2R*Eo*sqroot(m/2Eo) > h/4*pi

sqroot(Eo) > h/4*pi*R*sqroot(2m)

here h is planks constant, m is mass of electron R is radius of the bubble

d. Phe = 0

then Pe = 4*sigma/R = 3Ek/4*pi*R^3

also, Ek = h/4*pi*R*sqroot(2m) = 3.9049*10^-20/R

3*3.9049*10^-20/4*pi*R^4 = 4*3.75*10^-4/R

R^3 = 6.214*10^-18 m^3

R = 1.836*10^-6 m

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