Assume the following problem:
Two parallel-plate capacitors with different distances between the plates are connected in parallel to a voltage source. A point positive charge is moved from a point 1 that is exactly in the middle between the plates of a capacitor $C_1$ to a point 2 (of a capacitor $C_2$) that lies at a distance from the negative plate of $C_2$ equal to half the distance between the plates of $C_1$. Is any work done in the process by the electric field?
My Approach: Since the motion of the charge is perpendicular to the electric field of the capacitors, the work done in such a case would be zero. Hence, no work is done in the process by the electric field
Textbook Approach: The textbook says that the potential at point 1 is lower than the potential at point 2, hence $W = Q(V_2-V_1) > 0$ or positive work is done in moving the charge from 1 to 2
Where am I going wrong here? I don't see any mistake in my application of concepts. Is the electric field of a capacitor not always parallel and would I also have to account for edge effects or something similar?