IPhO2011_Th_Q2

Theoretical Competition: 12 July 2011 Question 2 Page 1 of 2 2. An Electrified Soap Bubble A spherical soap bubble with internal air density i , temperature i T and radius 0 R is surrounded by air with density a , atmospheric pressure a P and temperature a T . The soap film has surface tension , density s and thickness t . The mass and the surface tension of the soap do not change with the temperature. Assume that 0 R t . The increase in energy, dE , that is needed to increase the surface area of a soap-air interface by dA , is given by dE dA where is the surface tension of the film. 2.1 Find the ratio i i a a T T in terms of , a P and 0 R . [1.7 point] 2.2 Find the numerical value of 1i i a a T T using 10.0250Nm , 0 1.00 cm ,R and 5 21.013 10 Nm a P . [0.4 point] 2.3 The bubble is initially formed with warmer air inside. Find the minimum numerical value of i T such that the bubble can float in still air. Use 300 K a T , -31000 kgm s , -31.30 kgm a , 100 nmt and 29.80 msg . [2.0 points] After the bubble is formed for a while, it will be in thermal equilibrium with the surrounding. This bubble in still air will naturally fall towards the ground. 2.4 Find the minimum velocity u of an updraught (air flowing upwards) that will keep the bubble from falling at thermal equilibrium. Give your answer in terms of 0 , , , s R g t and the air’s coefficient of viscosity . You may assume that the velocity is small such that Stokes’s law applies, and ignore the change in the radius when the temperature lowers to the equilibrium. The drag force from Stokes’ Law is 0 6F R u . [1.6points] 2.5 Calculate the numerical value for u using 5 1 11.8 10 kgm s . [0.4 point] The above calculations suggest that the terms involving the surface tension add very little to the accuracy of the result. In all of the questions below, you can neglect the surface tension terms. Theoretical Competition: 12 July 2011 Question 2 Page 2 of 2 2.6 If this spherical bubble is now electrified uniformly with a total charge q , find an equation describing the new radius 1 R in terms of 0 , , a R P q and the permittivity of free space 0 . [2.0points] 2.7 Assume that the total charge is not too large (i.e. 2 4 0 0 a q P R ) and the bubble only experiences a small increase in its radius, find R where 1 0R R R . Given that (1 ) 1nx nx where 1x . [0.7 point] 2.8 What must be the magnitude of this charge q in terms of 0 0 , , , , , a s a t R P in order that the bubble will float motionlessly in still air? Calculate also the numerical value of q . The permittivity of free space 12 0 8.85 10 farad/m . [1.2 point]