CAPACITORS and DIELECTRICS (INSULATORS)

A capacitor can be "charged" and can store charge. When a capacitor is being charged, negative charge is removed from one side of the capacitor and placed onto the other, leaving one side with a negative charge (-q) and the other side with a positive charge (+q). The net charge of the capacitor as a whole remains equal to zero. Any two conductors insulated from one another form a CAPACITOR.

CAPACITOR

The amount of charge that can be placed on a capacitor is proportional to the voltage pushing the charge onto the positive plate. The larger the potential difference (voltage) between the plates, the larger the charge on the plates: Q = C V  or  C = Q / V The constant of proportionality is called the "capacitance" and is proportional to the area (A) of one of the plates and inversely proportional to the separation between the plates (d):   C = e A / d for a parallel plate capacitor, where e is the permittivity of the insulating material (or DIELECTRIC) between the plates. Recall that we used Gauss's Law to calculate the magnitude of the electric field (E) between the plates of a charged capacitor: E = s / eo where the space between the plates is a vacuum. Vab = E d,  so   E = Vab / d a) A charged parallel plate capacitor. (b) When the plate separation is small compared to the plate area, the fringing of the electric field at the edges is negligible.

The unit of capacitance is called the Farad (F). One Farad is equal to one Coulomb per Volt. ( F = C / V )

Capacitors, like resistors, can be connected in a circuit in two ways, series or parallel (or combinations of series and parallel).
The equivalent capacitance of a series connection is  1 / C_eq = 1 / C₁ + 1/ C₂     (Just the opposite from resistors!)

(a) Two capacitors in series and   (b) The equivalent capacitor.

The equivalent capacitor of a parallel connection is  C_eq = C₁ + C₂  (Just the opposite from resistors!)

(a) Two capacitors in parallel and  (b) the equivalent circuit.

ENERGY STORAGE in CAPACITORS and ELECTRIC FIELD ENERGY

Capacitors can store charge and ENERGY. We can calculate the potential energy (U) stored in a charged capacitor. Since the charging of a capacitor can be thought of as moving charge from one plate DIRECTLY onto the other plate through a potential difference of V, the voltage between the capacitor plates. Energy = Work = q * V_ab, and the potential Vab increases as the charge is placed on the plates (V_ab = Q / C). Since the V_ab changes as the Q is increased, we actually have to integrate over all the little dq being added to a plate which leads eventually
to:
U =  (1/2) Q² / C = (1/2) C V² =  (1/2) Q V

So the energy stored in a capacitor can be thought of as the potential energy stored in the system of positive charges that are separated from the negative charges, much like a stretched spring has potential energy associated with it.

ELECTRIC FIELD ENERGY

Here's another way to think of the energy stored in a charged capacitor. If we consider the space between the plates to contain the energy (equal to 1/2 C V²) we can calculate an energy DENSITY (Joules per volume). The volume between the plates is Area x distance between plates, or A d. Then the energy density (u) is

u =  (1/2) C V² / A d  or  u = (1/2) e_o E²

where we have used C = e_o A / d and V =E d.

This is an important result because it tells us that  empty space can contain energy,  i.e., if there is an electric field in the "empty" space.

If we can get an electric field to travel (or propagate) through empty space we can send or transmit energy!!!

Fig. 24.12 Effect of a dielectric between the plates of a parallel plate capacitor. The electrometer measures the potential difference. (a) With a given charge, the potential difference is V_o.
(b) With the same charge but with a dielectric between the plates, the potential difference V is smaller than V_o.

Most capacitors have an insulating (or DIELECTRIC) material between the plates. The presence of this dielectric INCREASES the capacitance of the capacitor compared to when the space between the plates was empty (a vacuum). Here's why:. The dielectric constant (K) is a positive number equal to 1 for a vacuum and greater than 1 for other dielectric materials (Teflon - 2.1, glass = 7, water = 80, etc.). When a dielectric is placed between plates, that the potential of a capacitor (V) decreases by a factor of K. However the Capacitance C = Q/V - thus when V is reduced but Q does not change then the capacitance  increases.

 (And since V = E d, we can conclude that the electric field between the plates must be reduced as a dielectric is inserted Q = (constant). See below for a physical picture of what is happening.)

 (a) Electric field lines with vacuum between the plates.
(b) The induced charges on the faces of the dielectric decrease the electric field. (See figure below for a physical picture of how the (blue) induced charges on the faces of the dielectric are established.)

The electric field between the capacitor plates is reduced by the presence of the dielectric because the induced surface charges on the dielectric (see figure below) cause an electric field in the opposite direction of the original field in the charged capacitor. These fields tend to cancel each other resulting in a reduction of the original field.

Molecules in the dielectric material have their positive and negative charges separated slightly, causing the molecules to be oriented slightly in the electric field of the charged capacitor. Polarization of a dielectric in an electric field gives rise to thin layers of bound charges on the surfaces, creating positive and negative surface charge densities. The sizes of the molecules are greatly exaggerated for clarity.

a) Electric filed of magnitude Eo between two charged plates. (b) Introduction of a dielectric of dielectric constant K. (c) the induced surface charges and their field (thinner lines). (d) Resultant field of magnitude Eo/K when a dielectric is between charged plates.