Two helium-filled, spherical balloons, each with charge q, are tied to a 7.80-g

Question

Two helium-filled, spherical balloons, each with charge q, are tied to a 7.80-g mass with strings of negligible mass, and the system floats in equilibrium as shown in the figure below. The distance between the balloons is L_1 = 50.0 cm, and the strings are L_2 = 110.0 cm long. Ignore the weight of the balloon material, and assume that the density of air is 1.29 kg/m^3 and the density of helium inside the balloons is 0.200 kg/m^3. Find the magnitude of the charge q on each balloon. Assume that the charge on each balloon acts as if it were concentrated at its center. Find the volume of each balloon.

Explanation / Answer

(a)

Apply Columbs law to solve the problem

F = k q1 q2/ r^2

from the figure

cos theta = sqrt ( 1.1)^2 - ( 0.25)^2 = 1.071

sin theta = 0.25/1.1 = 0.227

2 T cos theta = mg

T = mg/ 2 cos theta

= 7.8 * 10^-3 ( 9.8)/1.071

=0.0713 N

Coumb force = tension force

k q^2/ r^2 =T sin tehta

9 * 10^9 * q^2/0.5^2 = 0.0713 * 0.227

q= 0.67 * 10^-6 C

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(b)

U = m'g + T cos theta

V density( air) g = V desnity (ballon)g+ 7.8 * 10^-3 kg * g/2

V ( 1.29) g = V*0.2g + 3.9 * 10^-3 g

V (1.29-0.2) = 3.9 * 10^-3

V = 3.57 * 10^-3 m^3

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