How to prove that the electric field strength in the interlayer is radial?
This is a structure modified from a typical metal spherical capacitor: half of the interlayer is now filled with one kind of uniform, isotropic, and linear dielectric medium whose dielectric constant is $\epsilon$. See below: Given the radii of the ball inside and the exterior shell, now I have to calculate the capacity of this modified capacitor. My approach is that, as I use when calculating other capacities, let the ball be charged with positive charge $Q$, then calculate the electric field strength distribution in the interlayer, and finally calculate the integral $$\int_{\vec{R_1}}^{\vec{R_2}}\vec{E}(\vec{r})\centerdot d\vec{r}$$ to obtain the voltage decrease from the "anode" to the "cathode". Usually I will apply Gauss Theorem to a properly-selected close surface and work out the distribution (to be honest, maybe that's the only thing I can do to solve such problems). It is very easy but always depends terribly on the symmetry.
Now I am stuck here. 903
The yellow sector is the Gaussian surface I selected, concentric with the ball and with a tiny solid angle $d\Omega$ as is illustrated in my drawing. Now if I know that the electric strength in the interlayer, then the only thing I need to do is to calculate the $\vec{E}$ flux through the "bottom" and the "top" excluding the laterals, which is the key to the success of the application of Gauss Theorem here. But the prerequisite is that in the interlayer the field strength has only radial component, say, without tangent component. Were it not for the dielectric medium that breaks the symmetry this part would be just self-evident. But since the medium is here now, does it hold that the field strength is still radial and Gauss Theorem is still usable here? I believe yes, because the result is indeed rather simple and falls just within my expectation based on Gauss Theorem: $$C=\frac{2\pi (1+\epsilon) R_1 R_2}{R_2-R_1}$$ But I just can't explain why. Or, perhaps the electric field strength is no longer radial in this case (but I deem it much less possible), then with what else can we obtain the result? 903