why is it physically reasonable to expect that two identical parallel plate capacitors that are placed in series ought to have half the capacitance as one capacitor
In series
the charge has to be the same on both, call it Q
by definition each capacitance C= Q/V
so Veach = Q/C
now in series the voltages on each capacitor add so
V total = 2 Q/C
but by definition our equivalent capacitance = Q/Vtotal = C/2
Use the same argument but same voltage on each
To understand why two identical parallel plate capacitors in series have half the capacitance of a single capacitor, let's first break down the concept.
A capacitor is a device that stores electrical energy in the form of an electric field, typically made of two conductive plates separated by a dielectric material. The capacitance of a capacitor represents its ability to store charge and is measured in farads (F).
When capacitors are connected in series, they are arranged one after another, such that the positive plate of one capacitor is connected to the negative plate of the next capacitor. The total capacitance of capacitors in series can be calculated using the following formula:
1/C_total = 1/C1 + 1/C2 + 1/C3 + ...
Where C_total is the total capacitance and C1, C2, C3, etc., represent the individual capacitance values of the capacitors.
Now, let's consider two identical parallel plate capacitors in series. Since they are identical, their capacitance values will be the same, let's call it C.
Using the formula, we can calculate the total capacitance:
1/C_total = 1/C + 1/C
Simplifying the equation:
1/C_total = 2/C
Inverting both sides:
C_total = C/2
From this equation, it is clear that the total capacitance C_total is equal to half of the individual capacitance C.
Physically, this can be understood by considering how the electric fields across the parallel plates of the capacitors interact when they are connected in series. Since the capacitors share the same electric charge, the overall effect is that the same amount of charge will be stored at each parallel plate of the two capacitors. As the capacitance represents the ability to store charge, it is reasonable to expect that the total capacitance decreases when the charge is shared across two identical capacitors.
So, when two identical parallel plate capacitors are placed in series, the total capacitance is reduced to half of the capacitance of a single capacitor due to the sharing of electric charge between the plates.