Lecture 4 Capacitance and Capacitors Chapter 16.6  Outline Definition of Capacitance Simple Capacitors Combinations of Capacitors Capacitors with Dielectrics

Capacitance Introduction The capacitance C of a capacitor is the ratio of the charge (Q) on either conductor plate to the potential difference (  V) between the plates. Q C    V General definition Units of capacitance are farads (F) 1F  1C/1V C(Earth) ~ 1 F C(adult) ~ 150 pF =  12 F History

The Parallel-Plate Capacitor The capacitance of a parallel-plate capacitor whose plates are separated by air is: A C = є 0  d A is the area of one of the plates d is the distance between the plates є 0 is permittivity of free space More about capacitors

Capacitors Problem: A parallel-plane capacitor has an area of A=5cm 2 and a plate separation of d=5mm. Find its capacitance. Unit conversion: A = 5 cm 2 = 5 10  4 m 2 d = 5 mm = 5 10  3 m C = є 0 A/d =  12 C 2 / (N m 2 ) 5 10  4 m 2 / 5 10  3 m =  13 C 2 /(N m)=  13 F = pF N/C = V/m  C/N = m/V, F=C/V C 2 /(N m)=C (C/N)/m = C (m/V)/m = C/V = F

Combinations of Capacitors In real electric circuits capacitors can be connected in various ways. In order to design a circuit with desired capacitance, equivalent capacitance of certain combinations of capacitors can be calculated. There are 2 typical combinations of capacitors: Parallel combination Series combination

Parallel Combination

The left plate of each capacitor is connected to the positive terminal of a battery by a wire  the left plates are at the same potential  the potential differences across the capacitors are the same, equal to the voltage of the battery (  V). The charge flow ceases when the voltage across the capacitors equals to that of the battery and the capacitors reach their maximum charge. Q = Q 1 + Q 2 Q 1 = C 1  V Q 2 = C 2  V Q = C eq  V C eq  V = C 1  V + C 2  V C eq = C 1 + C 2 Examples

Series Combination

The magnitude of the charge is the same on all the plates. The equivalent capacitor must have a charge –Q on the right plate and +Q on the left plate. Q  V =  C eq  V =  V 1 +  V 2  V 1 = Q/C 1  V 2 = Q/C 2 Q Q Q  =  +  C eq C 1 C  =  +  C eq C 1 C 2 Examples

Energy Stored in a Capacitor The work required to move a charge  Q through a potential difference  V is  W =  V  Q.  V = Q/C, Q is the total charge on the capacitor. The voltage on the capacitor linearly increases with the magnitude of the charge. Additional work increases the energy stored. W = ½ Q  V = ½ (C  V)  V = ½C (  V) 2 = Q 2 /2C

Capacitors with Dielectrics A dielectric is an insulating material. The dielectric filling the space between the plates completely increases the capacitance by the factor  > 1, called the dielectric constant. If  V 0 is the potential difference (voltage) across a capacitor of a capacitance C 0 and a charge Q 0 in the absence of a dielectric. Filling the capacitor with a dielectric reduces the voltage by the factor  to  V, so that  V =  V 0 / . C = Q 0 /  V = Q 0 /  V 0 /  =  Q 0 /  V 0 =  C 0

Dielectric Strength For a parallel-plate capacitor: C =  є 0 A/d The formula shows that the capacitance can be made very large by decreasing the plate separation. In practice, the lowest value of d is limited by the electric discharge through the dielectric. The discharge occurs when the electric field in the dielectric material reaches its maximum, called dielectric strength. Dielectric strength of air is V/m.

Summary Capacitance is defined as the charge over the potential difference Capacitance of parallel-plate capacitor is directly proportional to the plate area and inversely proportional to the plate separation The equivalent capacitance of a parallel combination of capacitors equals to the sum of individual capacitances The inverse equivalent capacitance of a series combination of capacitors equals to the sum of the inverse individual capacitances Placing a dielectric between the plates of a capacitor increases the capacitance by a factor , called the dielectric constant