CAPACITANCE A capacitor is a device that can store electric charge, and consists of two conducting objects placed near each other but not touching. Capacitors are widely used in electronic circuits. They store charge which can later be released as in a camera flash, and as energy backup in computers if the power fails. Capacitors block surges of charge and energy to protect circuits. Very thin capacitors serve as memory for the ones and zeroes of the binary code in the random access memory (RAM) of computers. • For a given capacitor, it is found that the amount of charge Q acquired by each plate is proportional to the magnitude of the potential difference V between them. • Q = C V The constant of proportionality, C is called the capacitance of the capacitor. The unit of capacitance is coulombs per volt, and this unit is called a farad (F). Common capacitors have capacitance in the range of 1pF ( pico farad = 10-12F ) to 103μ F (microfarad = 10-6F ). The capacitance C does not in general depend on Q or V. Its value depends only on the size, shape, and relative position of the two conductors and also on the material that separates them. For a parallel plate capacitor whose plates have area A and are separated by a distance d of air, the capacitance is given by C = Ԑo A/d • We see that C depends only on geometric factors, A and d , and not on Q or V. The constant Ԑo is the permittivity of free space, which has the value 8.85 X 10-12 C2/ N m2 Capacitors in Parallel When a potential difference V is applied across several capacitors connected in parallel, that potential difference V is applied across each capacitor. The total charge q stored on the capacitors is the sum of the charges stored on all the capacitors. Capacitors connected in parallel can be replaced with an equivalent capacitor that has the same total charge q and the same potential difference V a the actual capacitors. q1 = C1V q2 = C2V q3V Q = q1 + q2 + q3 = Ceq V C1V + C2V + C3V = CeqV Ceq = C1 + C2 + C3 When a potential difference V is applied across several capacitors connected in series, the capacitors have identical charge q. The sum of the potential differences across all the capacitors is equal to the applied potential difference V. Capacitors that are connected in series can be replaced with an equivalent capacitor that has the same charge q and the same total potential difference V as the actual series capacitance q V V1 V2 V3 Ceq q q q q C1 C2 C3 Ceq 1 1 1 1 Ceq C1 C2 C3 Example 1. Capacitor calculations. a) Calculate the capacitance of a parallel plate capacitor whose plates are 20 cm X 3 cm and are separated by a 1 mm air gap. b) What is the charge on each plate if a 12V battery is connected across the two plates? C) What is the electric field between the plates? d) Estimate the area of the plates needed to achieve a capacitance of 1F, given the air gap d. Solution: a) The area A = (20 X 10-2 m) ( 3 X 10-2 m ) = 6 X 10-3 m2 The capacitance is then: C = Ԑo A/d = 8.85 X 10-12 C2/ N m2 ( 6X10-3 m2 / 1 X 10-3 m = 53 Pf b) The charge on each plate is: Q = CV = ( 53 X 1012F) ( 12V) = 6.4 X 10-10 C c) E = V / d = 12V / 1 X 10-3 m = 1.2 X 104 V /m d) A = Cd / Ԑo = ( 1F ) ( 1 X 10-3 m ) / 9 X 10-12 C2 / N m2 ) = 108 m2 • DIELECTRICS In most capacitors there is an insulating sheet of material, such as a paper or plastic called a dielectric between the plates. This serves several purposes. First, dielectrics do not break down as readily as air, so higher voltages can be applied without charge passing across the gap. Furthermore, dielectric allows the plates to be placed closer together without touching, thus allowing an increased capacitance because d is less. Thirdly, it is found experimentally that if the dielectric fills the space between the two conductors, it increases the capacitance by a factor K, known as the dielectric constant. Thus for a parallel-plate capacitor: C = K Ԑo (A/d) this can also be written as C = Ԑ A / d where Ԑ K Ԑo is called the permeability of the material. The potential difference between the plates, Vo is given by Q = Co Vo Storage of Electrical Energy A charged capacitor stores electrical energy by separating positive and negative charges. The energy stored in a capacitor will be equal to the work done to charge it. The net effect of charging a capacitor is to remove charge from one plate and add it to the other plate. This is what the battery does when it is connected to a capacitor. Initially, when the capacitor is uncharged, no work is required to move the first bit of charge over. As more charge is transferred, work is needed to move charge against increasing voltage V. The work needed to add a small amount of charge ∆q, when a potential difference V is across the plates is ∆W = - V ∆q. The total work needed to move total charge Q is equivalent to moving all the charge Q across a voltage equal to the average voltage during the process. The average voltage is ( Vf 0 ) / 2 = Vf / 2 where Vf is the final voltage, so the work to move the total voltage Q from one plate to the other is: W = Q Vf / 2 Thus we can say that the electric potential energy, PE, stored in a capacitor is PE = energy = ½ Q Vf where V is the potential difference between the plates and Q is the charge on each plate. Since Q = CV we can also write PE = ½ QV = ½ CV2 = ½ Q2 / C Example 1. Energy Stored in Capacitor. A camera flash unit stores energy in a 150μF capacitor at 200 V. How much electric energy can be stored? Solution: PE = ½ CV2 = ½ ( 150 X 10-6F) ( 200 V )2 = 3 J