## PARALLEL-PLATE CAPACITORS:

 eo = Permittivity of empty space (e air ~ eo) = 8.854x10-12 C 2/N.m2 = 1/(4p k) e = Permittivity of material between plates A = Surface Area of one of the plates (SI: m 2) d = Separation of the plates (SI: m)

Electric Field in a Ideal Parallel Plate Capacitor:

• When two plate of different charge are placed near each other, the two E-fields between the plates add while the E-field outside the plate cancel.

• When the plates are close to each other to form a capacitor, the E-field between the plates is constant through out the interior of the capacitor as long as one is not near the edges of the plates.
• Since the electric field is the negative of the gradient of the potential and the E-field is constant inside a capacitor, the magnitude of the Electric field has a very simple relation to the voltage between the plates and their separation d.

• This relationship is also true in an electrical wire were V is the voltage across the ends of the wire and d is the length of the wire.

• Using the definition of capacitance we can determine the capacitance C of an ideal capacitor as a function of its structure.

• This equation for the capacitance of a parallel capacitor shows that C is a constant independent of the charge stored in on the plates or the voltage across the capacitor.

• By placing a thin insulating material (a dielectric) between the plates the separation d can be reduced thus increasing the capacitance of the capacitor and prevent the plates from touching.

• It takes more voltage to store the same amount of charge on a capacitor because of the presence of the dielectric. Typically a dielectric contains polar molecules which partially line up in the presence of the electric field. The dielectric creates an E-field in the opposite direction which reduces the overall E-field between the plates.

• You could calculate the capacitance C of a parallel plate capacitor by replacing the permittivity of empty space eo by the permittivity e of the dielectric material placed between the plates.

• Except for simple capacitors (like the parallel plate capacitor) we do have an equation from which we can calculate the capacitance C. In practace we simply measure the value of a capacitor and assume that it is constant.

Energy Density of the Electric Field in a Capacitor:

• The electrical energy stored in the Electric Field between the plates of an ideal capacitor has a simple form when expressed as the electrical energy per unit volume, u = U/Vol

• This is a general expression that is valid for the energy density of the Electric Field no matter how the electric field is generated, i.e. it is true at any point in space where there is an electric field E.

• For parallel plate capacitors this can easily be derived since the E-field is constant through out the interior of the capacitor and equal to V/d. Here,