Gauss' Law has shown us that the electric field between two parallel plates

 

 

can be calculated with the formula

 

 

 

If the strength of the electric field between the plates becomes too strong, then the air between them can no longer insulate the charges from sparking, or discharging, between the plates. For air, this breakdown occurs when the electric field is greater than 3 x 10⁶ V/m. In order to keep this from happening, an insulator, or dielectric, is often inserted between the plates to reduce the strength of the electric field, without having to reduce the voltage being placed across the plates.

 

A dielectric is a polar material whose electric field aligns to oppose the original electric field already established between the plates. The dielectric is measured in terms of a dimensionless constant, κ ≥ 1, whose value is usually referenced from a table.

 

Material   κ air   1.00054 ethanol   24.3 glass   5-10 mica   3-6 paper   2-4 paraffin   2.1-2.5 polystyrene   2.3-2.6 porcelain   5.7 rubber   2-3 teflon   2.1 water   80

 

If this insulating material is insufficient then the capacitor can still leak allowing current to flow between the plates. When this occurs the electric device "smells as if something is burning."

 

κ = Eoriginal / Edielectric κ = Eo / Ed κ = Cdielectric / Coriginal κ = Cd / Co   When the battery is removed, the dielectric will decrease the electric field strength and the voltage between the plates while it increases their capacitance.   E = V/d

 

Using the fact that V = Ed and that capacitance is the ratio of the charge stored per unit volt we derived the following formula for the capacitance based on the geometry of a parallel-plate capacitor.

 

 

For more practice with capacitors and both connected and disconnected batteries, reference this worksheet.

 

 

 

Often capacitors can have complicated dielectric configurations based on available materials and circuit requirements. The good news is that they can be solved based on the principles of capacitors in series and in parallel, in combination with the formulas for the electric field and the geometry of a parallel plate capacitor.

 

If the capacitors are arranged in series (one after another along a single path), then

 

C_series = (1/C₁ + 1/C₂ + 1/C₃)^-1

 

If the capacitors arranged in parallel (strung along multiple paths that cross the same section), then

 

C_parallel = C₁ + C₂ + C₃

 

Let's work a few examples.