A Van de Graaff machine has a dome of radius 2.0 m. A motor causes a charged ins

Question

A Van de Graaff machine has a dome of radius 2.0 m. A motor causes a charged insulating belt to travel continuously between grounded spikes at the bottom of the device and another set of spikes connected to the dome at the top. Charge is transferred through the upper spikes to the outer surface of the dome at a rate of 17 microCoulombs per second. How long does it take to charge the dome to 3.0 MV? How much power is required to operate the machine when the dome potential is 3.0 MV, assuming the rate of charge transfer remains constant?

Explanation / Answer

From Gauss' Law, any spherical distribution of charge with outer radius R centered on the origin has an E-field in the region r > R identical to that of a point charge at the origin. So the voltage equation for a Van de Graaff generator in the region outside the dome, r > R, is the same as that for a point charge with the same total charge.

V = kq/r

==> q = r V/k = 2*3e6/9e9 = 6.6667e-4 C

time = t = 6.6667e-4/17e-6 = 39.2

b)

transfered energy per second= q V = 17e-6 * 3e6 = 51 W

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