So as I was doing my sample physics midterm, I realized a paradox or a concept that I just haven't fully grasped, hence I see a problem. 1. The problem statement, all variables and given/known data Suppose we have two parallel plates of a capacitor with: capacitance C, area A, distance of separation D, a constant voltage V=10V supplied by the battery that is connected to the capacitor, electric field E=V/d, surface charge density σ=(E*ε0) As we insert a piece of dielectric wth κ=1275 WHILE THE CAPACITOR REMAINS CONNECTED TO THE 10V battery, find the NEW: Electric field (E1), charge density σ1, Capacitance (C1), Potential Energy U. 3. The attempt at a solution So here is the paradox. 1. Since capacitor is still connected to the battery, V across capacitor must remain constant, so V1=V=10V. 2. If V is constant, and C increases due to the insertion of the dielectric, then total Q must increase. 3. if total Q increases, then the charge density must increase since A is the same. if σ (charge density) increases, then the new electric field must also increase, since E1=σ1/ε 4. If the electric field increases, the new voltage is given by V1=E1D. D remains the same, E1 increases, so the new voltage increases. However, we were just told that the voltage is held constant at 10V, so how can it increase and stay at 10V at the same time? This does not make sense to me! Please explain! Many thanks.