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: E_crit = 7.8 x 10³statvolts/m

All I have is:

 E _{air ll} = E _{oil ll}E _{air _l_} = 3 E _{oil _l_}

So  E_critll² + 9 E_{crit_l_}² = 1 x 10⁸ statvolts²/m²???