A van de Graaf generator has a motor that runs its 39 cm long (full length aroun

Question

A van de Graaf generator has a motor that runs its 39 cm long (full length around) rubber belt at a speed of 129 cm/s. When the motor has run for 12 s, the safety pole bends like a spring of stiffness 2.6 N/m, pulling its ball-like tip inward to a distance of 53 cm from the center of the tip of the safety pole to the center of the 23 cm diameter globe. They have an unbent distance of 56 cm.

(1) What is the charge on the small ball at t = 12 s? _______ C

(2) What is the magnitude of the electric field that pulls the small ball toward the large one? ________ kN/C

(3) What is the magnitude of the electric field half way between the two balls? ________ kN/C

(4) What is the surface charge density of the sphere? ________ C/m2

(5) What is the linear charge density on the charged part of the belt? _______ nC/m

(6) What is the total charge on the belt at any given time? _________ nC

Explanation / Answer

1. uncharged distaqnce between the pole and the ball, d1 = 56 cm

after getting charged, distance, d2 = 53 cm

expansion in spring = x = d1 - d2 = 0.03 cm

so force due to spring, F = kx = 2.6*0.03 = 0.078 N [ k = 2.6 N/m given]

so for equal charge on both pole and the mass = q

F = kq^2/d2^2 = 0.078 N

8.98*10^9*q^2/0.53^2 = 0.078

q = 1.562*10^-6 C

2. Magnitude of electric field = F/q = 0.078/1.562*10^-6 = 49935.522 N/C

3. electric field halfway be E

then E = kq/(0.53/2)^2 + kq/(0.53/2)^2 = 4*kq/0.53^2 = 199740.263 N/C

4. surface charge density of sphere = q/4*pi*r^2 [ where r is radius of sphere r= 0.23/2 m]

sigma = 1.562*10^-6 / pi*0.23^2 = 9.3988*10^-6 C/m^2

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