Lecture #8 Electric dipoles. Dielectrics. Electric Capacitance and Capacitors

Capacitors connected in series
Conservation of energy (Kirchhoff's voltage rule) requires that the sum of the individual capacitor voltages be equal to the total voltage supplied by the battery:
Capacitors connected in series
V = V1 + V2 + ... + Vn
A second addition rule for capacitors
The reciprocal of the equivalent capacitance of two capacitors connected in series is the sum of the reciprocals of the individual capacitances
For capacitors connected in parallel, the voltage is the same across each capacitor, while for capacitors connected in series, the charge is the same on each capacitor.

4.Energy Storage in a Capacitor

A test charge q placed in this constant field experiences a force F = qE.
If the charge moves through the distance d from the positive to the negative plate, the work done by the field is
W = Fd = qEd.
This amount of work is equal to the loss of electric potential energy by the test charge.

Energy Storage in a Capacitor

The energy stored on the capacitor may be considered as electrical potential energy, which can be released when the capacitor is discharged by connecting a conducting path between its plates.
We can think of capacitors as devices for storing energy or as devices for storing electric charge.
The total work done in charging the capacitor from zero to a voltage V is the sum of the work needed to transfer each amount of charge Δq:

Energy Storage in a Capacitor

The energy required to charge an initially uncharged capacitor to a potential difference V and charge q is:
We have seen that, when fringing can be ignored, the electric field within a parallel-plate capacitor is uniform; that is, it is the same at all points between the plates. Thus the energy density u, the stored energy per unit volume, should also be uniform.
where Ad is the volume between the plates.
If we substitute the relationships C = κεoA/d and E = V/d the energy density becomes