Capacitance Electric Capacitance

Transcription

1 Dr. Alain Brizard College Physics II (PY 211) Capacitance Textbook Reference: Chapter 24 sections 1-5. Electric Capacitance A capacitor is an electrical device designed to store electric charge and, therefore, to store electric energy. The device is constructed by placing two charged conducting surfaces near each other (without touching). By design, the two surfaces (labeled a and b) carry charges q a = Q and q b = Q (so that the net charge accumulated on the two surfaces is zero) and a potential difference V ba = V b V a exists between the two surfaces. By definition, the capacitance C of a capacitor to store charge Q in the presence of a potential difference V ba is C = Q (SI units : F = C/V). V ba One of the most important application of capacitors is to rapidly deliver an electrical discharge. For example, a medical defibrillator is composed of two paddles to be placed on the chest of a patient (see Figure below). After a potential difference V ba is established between the paddles, charge Q = CV is made to flow (as a result of an electric field E) through the body (hopefully through the patient s heart), where C denotes the capacitance of the defibrillator. 1

2 Calculation of Capacitance The simplest capacitor is the parallel-plate capacitor constructed by placing two conducting planes (of equal area A) separated by a distance d (see Figure below); here, surface a (located at x = d) has potential V a and carries charge Q while surface b (located at x = 0) has potential V b >V a and carries charge Q. With charge Q accumulated on the capacitor plates, the electric field is uniform between the two plates and is expressed as E = 2 σ x = σ x, 2 ɛ 0 ɛ 0 where σ = Q/A denotes the surface-charge density. By definition, the potential difference V ba is expressed as 0 σ V ba = V b V a = dx = σd. d ɛ 0 ɛ 0 Hence, the capacitance of a parallel-plate capacitor is defined as C = Q V ba = Qɛ 0 σd = ɛ 0 where ɛ 0 =8.85 pf m 1 (pf = picofarad = F). For example, a parallel-plate capacitor made of two 1-m 2 conducting parallel plates placed at a distance of 1 mm has a capacitance of C = ɛ 0 (1 m 2 /10 3 m) = 8.85 nf and, when exposed to a potential difference of 100 V, it accumulates Q = C V = µc. As a second example, we consider the spherical capacitor constructed by placing two concentric conducting shells of radii b < a(see Figure below). Here, surface a (located 2 A d,

3 at r = a) has potential V a and carries charge Q while surface b (located at r = b) has potential V b >V a and carries charge Q. With charge Q accumulated on the capacitor shells, the electric field is expressed as Q E = 4πɛ 0 r r, 2 while it is zero inside the inner shell and outside the outer shell (from Gauss s Law). The potential difference V ba is expressed as b Qdr V ba = V b V a = a 4πɛ 0 r = Q ( 1 2 4πɛ 0 b 1 ), a and, hence, the capacitance of a parallel-plate capacitor is defined as C = Q ( 1 = 4πɛ 0 V ba b 1 ) 1 ( ) 4πab = ɛ 0. a a b We note from these two examples that expressions for capacitance always have ɛ 0 (with unit pf m 1 ) in their numerator multiplying a geometric factor (with unit m). Capacitors in Series and Parallel Capacitors can be added either in series or in parallel. Capacitors are added together either to decrease capacitance or (more often) to increase capacitance. When capacitors (C 1,C 2, ) are added in series, the same charge Q must be stored on each capacitor C i and, thus, the i th capacitor experiences a potential difference V i = Q/C i. 3

4 If the total potential difference is V = i V i, we can define the equivalent capacitance C eq (ser) as ( ) C eq (ser) Q 1 = i (Q/C i ) = 1. i C i When capacitors (C 1,C 2, ) are added in parallel, on the other hand, the same potential difference V must exist across each capacitor C i and, thus, the i th capacitor accumulates charge Q i = C i V on it. If the total accumulated charge is Q = i Q i, we can define the equivalent capacitance C eq (par) as C (par) eq = i Q i V = i Hence, capacitors are connected in parallel in order to increase capacitance while they are connected in series to decrease capacitance. When N identical capacitors with capacitance C are connected to each other, an equivalent capacitance of C/N results when they are connected or NC when they are connected in parallel. A hybrid connection is obtained when various capacitors are connected in series and parallel within a circuit; a hybrid connection of N identical capacitors has an equivalent capacitance C/N < C eq <NC. For a hybrid connection of three identical capacitors (a capacitor connected in series with a pair of two capacitors connected in parallel), for example, the equivalent capacitance is C i. C 3 < C eq = ( 1 C + 1 2C ) 1 = 2 C 3 < 3 C. 4

5 Electric Energy Storage Capacitor banks provide the most efficient way to store a large amount of electrical energy (> kj) and release it rapidly over a very short period of time (< µsec) and, thus, delivering high-power electrical pulses (> GW). The picture below (taken from Sandia National Laboratories) shows an electrical discharge (called a Z pinch). The pulse that drives Z lasts less than ten billionths of a second 20,000 times faster than a lightning bolt and yet carries 1,000 times the electrical current in a typical lightning bolt. 1 To obtain an expression for the electrical energy stored in a capacitor (with capacitance C), we begin with the infinitesimal work dw = Vdq = (q/c) dq needed to accumulate an infinitesimal charge dq when a potential difference V exists between the capacitor plates. It is quite clear that it should be increasingly difficult to add charge to a capacitor (since work must be done against the electric field). The total work done to accumulate charge Q on the capacitor is, therefore, expressed as Q W = C 1 qdq = Q2 0 2C, and, hence, the electrical energy stored in the capacitor is U = Q2 2C = 1 2 CV2 1 = 1 2 QV, 5

6 expressed in terms of Q, C, and/or V. Dielectrics An easy way to increase the capacitance of a capacitor is to fill the space between the capacitor plates with a dielectric medium, which multiplies the permittivity of free space ɛ 0 by a coefficient K>1 (known as the dielectric constant), such that the new capacitance is C = KC 0, where C 0 denotes the capacitance without the dielectric medium. A second advantage associated with the presence of a dielectric medium is associated with the fact that, since electrical breakdown occurs at higher electric fields in a dielectric medium, a larger potential difference can be applied before spontaneous electrical discharge occurs. Note that the stored electrical energy also increases (at constant potential difference V ) since U = KU 0. On the other hand, a smaller potential difference is needed to accumulate a fixed charge Q since V = V 0 /K and, thus, a smaller electric field E = E 0 /K is set up between the capacitor plates. 6