a conducting sphere of radius r at potential zero


A conducting sphere of radius R at potential zero is surrounded by a concentric spherical shell of dielectric material of inner radius R, outer radius 2R, and dielectric constant ε=7/5. Suppose that the sphere and the dielectric shell are placed in an initially uniform electric field E0 in the z direction.

Show that the potential is given by:

Φ(r,θ)=-E0rcosθ + (2E0R3/r2)cosθ    for r≥2R      

Φ(r,θ)=-(6/7)E0(r - R3/r2)cosθ           for  R≤r≤2R

It should be (ε2 - ε1)/(ε2 + 2ε1) ,  but if ε1=7/5  & ε2=1  then it's -2/19.  If I switch them it's 2/17. Why is the answer just 2?

Within a large volume of transformer oil of dielectric constant ε=3, there is a spherical air bubble. What is the maximum permissible strength of the original electric field in the transformer oil if the air in the bubble is not to suffer electric breakdown? The critical strength for electrical breakdown in air is 3 x 106 volts/m, or 1 x 104statvolts/m. Qualitatively, how does your answer change if the bubble is elongated in the direction of the electric field? If the bubble is flattened?

ANS: Ecrit = 7.8 x 103statvolts/m

All I have is:

 E air ll = E oil llE air _l_ = 3 E oil _l_

So  Ecritll2 + 9 Ecrit_l_2 = 1 x 108 statvolts2/m2???

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Electrical Engineering: a conducting sphere of radius r at potential zero
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