Consider two conductors carrying charges of equal magnitude but of opposite sign, Such a combination of two conductors is called a capacitor. The capacitance C of a capacitor is the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between them: Fig 26-CO All of these devices are capacitors, which store electric charge and energy. A capacitor is one type of circuit element that we can combine with others to make electric circuits. (Paul Silverman/Fundamental Photographs) The SI unit of capacitance is the farad (F) = coulombs per volt, Fig 26-CO, p.795

26.2 CALCULATING CAPACITANCE Parallel-Plate Capacitors A parallel-plate capacitor consists of two parallel conducting plates, each of area A, separated by a distance d. When the capacitor is charged by connecting the plates to the terminals of a battery, the plates carry equal amounts of charge. One plate carries positive charge +Q, and the other carries negative charge -Q. The value of the electric field between the plates is

Figure 26.3 (a) The electric field between the plates of a parallel-plate capacitor is uniform near the center but nonuniform near the edges. The capacitance of a parallel-plate capacitor is proportional to the area of its plates and inversely proportional to the plate separation Fig 26-3a, p.799

C = 8.85 x 10-12 (C2/N.m2) . 2x 10-4(m2)/ 1x 10-3 (m) Example A parallel-plate capacitor has an area A = 2.00 x 104 m2 and a plate separation d = 1.00 mm. Find its capacitance. C = 8.85 x 10-12 (C2/N.m2) . 2x 10-4(m2)/ 1x 10-3 (m) 1.77 x 10-12 F = 1.77 pF Figure 26.3 (b) Electric field pattern of two oppositely charged conducting parallel plates. Small pieces of thread on an oil surface align with the electric field. Fig 26-3b, p.799

2. The Cylindrical Capacitor A cylindrical capacitor consists of a solid cylindrical conductor of radius a and length surrounded by a coaxial cylindrical shell of radius b. Figure 26.6 (Example 26.2) (a) A cylindrical capacitor consists of a solid cylindrical conductor of radius a and length l surrounded by a coaxial cylindrical shell of radius b. (b) End view. The electric field lines are radial. The dashed line represents the end of the cylindrical gaussian surface of radius r and length l. Fig 26-6, p.801

3.The Spherical Capacitor A spherical capacitor consists of an inner sphere of radius a surrounded by a concentric spherical shell of radius b.

Other different capacitors shape

Ceq = C1 + C2 Ceq  V= C1  V+ C2  V Q= Q1+Q2 Q2= C2  V Q1= C1  V 26.3 COMBINATIONS OF CAPACITORS Parallel Combination Let us call the maximum charges on the two capacitors Q 1 and Q 2 . The total charge Q stored by the two capacitors is Q= Q1+Q2 Q2= C2  V Q1= C1  V The equivalent capacitor Q= Ceq  V Ceq  V= C1  V+ C2  V Ceq = C1 + C2

Active Figure 26.9 (b) The circuit diagram for the parallel combination. Fig 26-9b, p.803

Series Combination Fig 26-10, p.804 Active Figure 26.10 (a) A series combination of two capacitors. The charges on the two capacitors are the same. (b) The circuit diagram for the series combination. (c) The equivalent capacitance can be calculated from the relationship 1 /Ceq =1 /C 1+ 1/ C 2 . Fig 26-10, p.804

Example : Find the equivalent capacitance between a and b for the combination of capacitors shown in Figure Figure 26.11 (Example 26.4) To find the equivalent capacitance of the capacitors in part (a), we reduce the various combinations in steps as indicated in parts (b), (c), and (d), using the series and parallel rules described in the text. Fig 26-11, p.806

26.4 Energy stored in a charged capacitor Suppose that q is the charge on the capacitor at some instant during the charging process. At the same instant, the potential difference across the capacitor is V = q/C. We know that the work necessary to transfer an increment of charge dq from the plate carrying charge -q to the plate carrying charge +q (which is at the higher electric potential) is

This result applies to any capacitor, regardless of its geometry This result applies to any capacitor, regardless of its geometry. We see that for a given capacitance, the stored energy increases as the charge increases and as the potential difference increases

Energy stored in a parallel-plate capacitor For a parallel-plate capacitor, the potential difference is related to the electric field through the relationship V = Ed.. The capacitance is given by By substituting The energy per unit volume known as the energy density, is The energy density in any electric field is proportional to the square of the magnitude of the electric field at a given point

26.5. Capacitors with Dielectrics Dielectric is a non-conducting material, such as rubber, glass, or waxed paper. When a dielectric is inserted between the plates of a capacitor, the capacitance increases. If the dielectric completely fills the space between the plates, the capacitance increases by a dimensionless factor k , which is called the dielectric constant.

For a parallel-plate capacitor, we can express the capacitance when the capacitor is filled with a dielectric as We see that a dielectric provides the following advantages: • Increase in capacitance • Increase in maximum operating voltage • Possible mechanical support between the plates, which allows the plates to be close together without touching, thereby decreasing d and increasing C

Table 26-1, p.812