# Capacitor problem: What's wrong with this?

The capacitor shown contains 3 different dielectric material $$\epsilon_1,\epsilon_2,\epsilon_3$$ between the top and bottom metallic plates on which lie teh points $$A$$ and $$B$$ as shown. The electric field is oriented along the vertical direction (neglecting fringe effects). Because of the continuity of the electric field $$D$$ across the interface $$CD$$ we have $$\epsilon_1E_1=\epsilon_2E_2$$ and therefore $$E_1\ne E_2$$ for different dielectrics. But continuity of the tangential field across $$AB$$ (at different points) demands that $$E_1=E_3=E_2$$. What's wrong with it? My intuition is that the field cannot change abruptly on interfaces like $$AB$$ since it is basically generated by the contributions from surfaces charge area elements carrying charge density $$\sigma$$ and modified to take into account the polarization due to the dielectric - so they are actually $$\sigma/\epsilon_i$$. Each elements adds up a continuous field so the result of all the charges must be a field that varies continuously across $$AB$$ in some transition region. Since the only spatial dimension in the problem is the plate separation my guess is that this would also be the characteristic scale of the variation of the field. I would appreciate if anyone can confirm this.

• Is it legitimate to ignore fringe effects along ACB? – Bob D Dec 1 '19 at 19:06
• Probably not. I meant along the outside edges of the capacitor – am301 Dec 1 '19 at 19:17
• Yes I can understand that for edge D, but it would seem you can’t ignore it at A-C-B because that would definitely seem to be applicable to the transition to $e_3$. – Bob D Dec 1 '19 at 19:29