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Assume a spherical electrode of radius r1 and a spherical metal tank of radius r2. The electrode is inside the metal tank. The electrode must operate at a voltage V with respect to a tank. The tank is grounded (V=0). Now let's focus on the electric field intensity at the surface of the electrode, as that is where it is highest. And let's try to optimize r1 and r2.

a) If we disregard the cost of the tank, then its optimum radius will be that for which the electric field intensity in the insulating gas will be minimum, for this will permit using a minimum gas pressure. What is this optimum value of r2?

b) In practice, cost, weight and space limits r2. One must therefore optimize the ratio r2/r1 by varying r1. Show that the E value at the surface of the high voltage electrode (r=r1) has a minimum value of 2V/r1 when r2 = 2r1.

c) explain quantitatively why there should be an optimum condition.

d) to determine whether the optimum condition is critical or not, plot the value of E at r = r1 for an actual case where r2 was 0.483 meter and for values of r1 ranging from 0.1 to 0.4 meters.

e) what range of values of r1 can be tolerated if one can tolerate an electric field intensity 10% higher than 2V/r1 ?

f) Calculate 2V/r1 for V = 5 x 10^5.

related equations:

Coulomb's Law

Attempt:

For part one, I tried to use Coulomb's law.

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# Homework Help: Optimizing radius for minimus electric field intensity

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