Optimizing radius for minimus electric field intensity

[E&M]

Homework Statement

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.

 

[gabbagabbahey]

Are you familiar with Laplace's equation? In this case, due to symmetry, the potential between the two shells will depend only on the distance from the center of the spheres (i.e. V=V(r) in between the electrode and the tank) and so you should be able to easily solve Laplace's equation to determine V(r) and then take the gradient to get \vec{E}(r) in between the electrode and the tank.

After you know \vec{E}(r), optimizing it should be straight forward.

 

[E&M]

I firstly calculated D for the r> r1 and then calculated E for r1<r< r2. Using that I calculated V taking integration of E.dl. The part I was not sure about from here was about optmiizing E but how I did it was since, we are supposed to minimize E, which is same as maximizing Capacitance for given Q, I calculated C and I got answer as r1= r2. I hope I am not totally wrong.

 

[tithai000]

\overline{}k \frac{1}{2} A^{01}_{02}