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,

Example 1 An electric potential of 5 V is applied across a capacitor of 20 x10-6 F, the electric charge Q acquires is

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

C = 8.85 x 10-12 (C2/N.m2) . 2x 10-4(m2)/ 1x 10-3 (m) Example 2 A parallel-plate capacitor has an area A = 2 x 10-4 m2 and a plate separation d = 1x10-3 m. 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 Example 3 The capacitance of two parallel plates with an area A and separation d is 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.

Example 4 The capacitance of a parallel – plate capacitors having area 2 x10-4 m2 of each plate and separating distance of 5 m equals to

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.

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

V=V1=V2 and Q= Q1+ Q2 Parallel Combination Active Figure 26.9 (b) The circuit diagram for the parallel combination. V=V1=V2 and Q= Q1+ Q2

Q=Q1=Q2 and V=V1+V2 Series Combination 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 . Q=Q1=Q2 and V=V1+V2

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

Example 6 Two capacitors C1=6 PF, and C2= 4 PF connected in parallel What is an equivalent capacitance C total= And when connected in series, what is an equivalent capacitance  C1 C2 a b  c1 c2 a b

Example: the equivalent capacitance between a and b is  c1 c2 a b if C1=C2 = 4 nF  C1 C2 C3 a b If C1 = 4  F C2 = 5  F C3 = 8  F

We see that a dielectric provides the following advantages: • Increase in capacitance • Increase in maximum operating voltage

Ex A parallel plate capacitor of geometrical dimensions of 2 cm by 2 cm separated by 2 mm thickness Of Mylar ( K= 3.2 ). Its capacitance is