Capacitance and Dielectrics 1
Definition of Capacitance A capacitor consists of two conductors separated by an insulator. The capacitance of a given capacitor depends on its geometry and on the material (called a dielectric) that separates the conductors. Capacitance Is a capacity The capacitance C of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between the conductors: 2
The SI unit of capacitance is the farad (F), which was named in honor of Michael Faraday: 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, and the other carries negative charge. Suppose that we have a capacitor rated at 4 pf. This rating means that the capacitor can store 4 pc of charge for each volt of potential difference between the two conductors. 3
Calculating Capacitance We can derive an expression for the capacitance of a pair of oppositely charged conductors in the following manner: assume a charge of magnitude Q, calculate the potential difference using the techniques described in the preceding chapter. We then use the expression C = Q/ V to evaluate the capacitance. The capacitance of a pair of conductors depends on the geometry of the conductors. 4
Parallel-Plate Capacitors the capacitance of a parallel-plate capacitor is proportional to the area of its plates and inversely proportional to the plate separation 5
A circuit consisting of a capacitor, a battery, and a switch. 6
Cylindrical Capacitors A cylindrical capacitor consists of a solid cylindrical conductor of radius a and length l surrounded by a coaxial cylindrical shell of radius b. 7
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. Using Gauss s law that the magnitude of the electric field of a cylindrical charge distribution having linear charge density λ is E = 2k e λ/r 8
Combinations of Capacitors Parallel Combination the individual potential differences across capacitors connected in parallel are the same and are equal to the potential difference applied across the combination. the total charge on capacitors connected in parallel is the sum of the charges on the individual capacitors. 9
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the equivalent capacitance of a parallel combination of capacitors is the algebraic sum of the individual capacitances and is greater than any of the individual capacitances. 11
Series Combination the charges on capacitors connected in series are the same. the total potential difference across any number of capacitors connected in series is the sum of the potential differences across the individual capacitors. 12
the inverse of the equivalent capacitance is the algebraic sum of the inverses of the individual capacitances and the equivalent capacitance of a series combination is always less than any individual capacitance in the combination. 13
Energy Stored in a Charged Capacitor A plot of potential difference versus charge for a capacitor is a straight line having a slope 1/C. The work required to move charge dq through the potential difference V existing at the time across the capacitor plates is given approximately by the area of the shaded rectangle. The total work required to charge the capacitor to a final charge Q is the triangular area under the straight line, 14
Capacitors with Dielectrics A dielectric is a nonconducting material, such as rubber, glass, or waxed paper. When a dielectric is inserted between the plates of a capacitor, the capacitance increases. A charged capacitor before and after insertion of a dielectric between the plates. The charge on the plates remains unchanged, but the potential difference decreases from V 0 to V = V 0 /κ. Thus, the capacitance increases from C 0 to κc 0. 15
the capacitance increases by the factorκwhen the dielectric completely fills the region between the plates. For a parallel-plate capacitor, we can express the capacitance when the capacitor is filled with a dielectric as 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. 16
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Types of Capacitors Three commercial capacitor designs. A tubular capacitor, whose plates are separated by paper and then rolled into a cylinder. A high-voltage capacitor consisting of many parallel plates separated by insulating oil. An electrolytic capacitor. 18
Quick quiz If you have ever tried to hang a picture or a mirror, you know it can be difficult to locate a wooden stud in which to anchor your nail or screw. A carpenter s stud-finder is basically a capacitor with its plates arranged side by side instead of facing one another, as shown in figure. When the device is moved over a stud, does the capacitance increase or decrease? 19
PR Sebuah kapasitor keping-paralel diisi dengan dua buah dielektrik seperti telihat pada gambar. Tunjukkan bahwa Sistem ini dapat dipandang sebagai dua kapasitor seluas A/2 yang terhubung secara paralel dan Kapasitansinya naik sebesar faktor (κ 1 + κ 2 )/2. A d κ 1 κ 2 20