# Chapter 8. Capacitors. Charging a capacitor

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1 Chapter 8 Capacitors You can store energy as potential energy by pulling a bowstring, stretching a spring, compressing a gas, or lifting a book. You can also store energy as potential energy in an electric field, and a capacitor is a device you can use to do exactly that. There is a capacitor in a portable battery-operated photoflash unit, for example. It accumulates charge relatively slowly during the charging process, building up an electric field as it does so. It holds this field and its energy until the energy is rapidly released during the flash. Charging a capacitor One way to charge a capacitor is to place it in an electric circuit with a battery. An electric circuit is a path through which charge can flow. A battery is a device that maintains a certain potential difference between its terminals (points at which charge can enter or leave the battery) by means of internal electrochemical reactions. In Fig. 8.1a, a battery B, a switch S, an uncharged capacitor C, and interconnecting wires form a circuit. The same circuit is shown in the schematic diagram of Fig. 8.1b, in which the symbols for a battery, switch, and a capacitor represent those devices. The battery maintains potential difference V between its terminals. The terminal of higher potential is labelled + and is often called the positive terminal; the terminal of lower potential is labelled - and is often called the negative terminal. 54

2 55 Figure 8.1: (a) Battery B, switch S, and plates h and l of capacitor C, connected in a circuit. (b) A schematic diagram with the circuit elements represented by their symbols. The circuit shown in Figs. 8.1a and b is said to be incomplete because switch S is open; that is, it does not electrically connect the wire attached to it. When the switch is closed, electrically connecting those wires, the circuit is complete and charge can then flow through the switch and the wires. The charge that can flow through a conductor, such as a wire, is that of electrons. When the circuit of Fig.8.1 is completed, electrons are driven through the wires by an electric field that the battery sets up in the wires. The field drives electrons from capacitor plate h to the positive terminal of the battery; thus plate h becomes positively charged. The field drives just as many electrons from the negative terminal of the battery to capacitor plate l; this plate l, gaining electrons, becomes negatively charged just as much as plate h becomes positively charged. The potential difference between the initially uncharged plates is zero. As the plates

3 56 CHAPTER 8. CAPACITORS become oppositely charged, that potential difference increases until it equals the potential difference V between the terminals of the battery. Then plate h and the positive terminal of the battery are at the same potential, and there is no longer an electric field in the wire between them. Similarly, plate l and the negative terminal reach the same potential and there is then no electric fied in the wire between them. Thus, with the field zero, there is no further drive of electrons. The capacitor is said to be fully charged, with a potential difference V and a charge Q, which are related by Equ Q = CV (8.1) Note: Q is the magnitude of the charge on one plate of the capacitor. C is the capacitance of a the capacitor and its value depends only on the geometry of the plates and not on their charge or potential difference. The capacitance is a measure of how much charge must be put on the plates to produce a certain potential difference between them: the greater the capacitance, the more charge is required. The SI unit of capacitance that follows from Equ. 8.1 is the coulomb per volt. This unit occurs so often that it is goven a special name, the farad (F). Capacitor Networks When there is a combination of capacitors in a circuit, we can sometimes replace that combination with an equivalent capacitor, that is, a single capacitor that has the same capacitance as the actual combination of capacitors. With such a replacement, we can simplify the circuit, affording easier solutions for unknown quantities of the circuit. Here we discuss two basic combinations of capacitors that allow such a replacement. Capacitors in Parallel Figure 8.2 shows three capacitors in parallel to a battery. They are connected in parallel because the terminals of the battery are effectively wired directly to the plates of each of the three capacitors. Because the battery maintains a potential difference V betwen its terminal, it applies the same potential difference V across each capacitor.

4 57 Figure 8.2: (a) Capacitors in parallel. Each is connected directly to the voltage source just as if it were all alone, and so the total capacitance in parallel is just the sum of the individual capacitances. (b) The equivalent capacitor has a large plate area and can therefore hold more charge than the individual capacitors. For the three capacitors we can write Q 1 = C 1 V, Q 2 = C 2 V, Q 3 = C 3 V, The total charge on the parallel combination is Q = Q 1 + Q 2 + Q 3 (8.2) Q = C 1 V +C 2 V +C 3 V (8.3)

5 58 CHAPTER 8. CAPACITORS Q = (C 1 +C 2 +C 3 )V (8.4) The equivalent capacitance C eq, with the same total charge Q and applied potential difference V as the combination, is then C eq = Q V = C 1 +C 2 +C 3 (8.5) Capacitors in Series Figure 8.3 shows three capacitors connected in series to a battery, which maintains a potential difference V across the left and right terminals of the series combination. This arrangement produces potential differences V 1, V 2, and V 3 across capacitors C 1, C 2, and C 3, respectively, such that V 1 +V 2 +V 3 = V. We seek the single capacitance C eq that is equivalent to this series combination and thus can replace the combination. When the battery is connected, each capacitor must have the same charge Q. This is true even though the three capacitors may be of different types and may have different capacitances. The potential difference across each capacitor: V 1 = Q C 1 (8.6) V 2 = Q C 2 (8.7) V 3 = Q C 3 (8.8) The potential difference across the entire series combination is then V = V 1 +V 2 +V 3 (8.9)

6 59 Figure 8.3: (a) Capacitors connected in series. The magnitude of the charge on each plate is Q. (b) The equivalent capacitor has a larger plate separation d. Series connections produce a total capacitance that is less than that of any of the individual capacitors. V = Q C 1 + Q C 2 + Q C 3 (8.10) The equivalent capacitance C eq, is then ( 1 V = Q ) C 1 C 2 C 3 (8.11) C eq = Q V = 1 ( 1C1 + 1 C2 + 1 C3 ) (8.12) 1 C eq = 1 C C C 3 (8.13)

7 60 CHAPTER 8. CAPACITORS Storing Energy in an Electric Field Once the charging of a capacitor has begun, the addition of electrons to the negative plate involves doing work against the repulsive forces of the electrons which are already there. Equally, the removal of electrons from the positive plate requires that work is done against the attractive forces of the positive charges on that plate. The work which is done is stored in the form of electrical potential energy. where W = the energy stored by a charged capacitor (J) Q = the charge on the plates (C) V = the PD across the plates (V ) C = the capacitance (F) W = Q2 2C = 1 2 CV 2 = 1 QV (8.14) 2 Figure 8.4: Energy stored in the large capacitor is used to preserve the memory of an electronic calculator when its batteries are charged. The Medical Defibrillator The ability of a capacitor to store potential energy is the basic of defibrillator devices, which are used by emergency medical teams to stop the fibrillation of heart attack victims. In the portable version, a battery charges a capacitor to a high potential difference, storing a large amount of energy in less than a minute. The battery maintains only a modest potential

8 61 difference; an electronic circuit repeatedly uses that potential difference to greatly increase the potential difference of the capacitor. The power, or rate of energy transfer, during this process is also modest. Conducting leads ( paddles ) are placed on the victim s chest. When a control switch is closed, the capacitor sends a portion of its stored energy from paddle to paddle through the victim. As an example, when a 70µF capacitor in a defibrillator is charged to 5000V, the energy stored in the capacitor W = 1 2 CV 2 = 1 2 ( ) ( ) = 875J (8.15) About 200J of this energy is sent through the victim during a pulse of about 2.0ms. The power of the pulse is P = W t which is much greater than the power of the battery itself. = 200 = 100kW (8.16) Figure 8.5: Automated external defibrillators are found in many public places. These portable units provide verbal instructions for use in the important first few minutes for a person suffering a cardiac attack.

9 62 CHAPTER 8. CAPACITORS Joining two charged capacitors Fig. 8.6 shows two charged capacitors whose capacitances are C 1 and C 2. On closing S, charge flows until the potential difference across each capacitor is the same. The final charge, potential difference and energy of each capacitor can be calculated by making use of: (i) there is no change in the total amount of charge, (ii) the two capacitors acquire the same potential difference, (iii) the capacitors are in parallel and therefore the capacitance, C, of the combination is given by C = C 1 +C 2. Figure 8.6: Joining two charged capacitors. It turns out that unless the intial potential differences of the capacitors are the same, the total energy stored by the capacitors decreases when they are joined together. Energy is dissipated as heat in the connecting wire when charge flows from one capacitor to the other, and this accounts for the decrease in stored energy. Charging and Discharging a Capacitor through a resistor Many electronic circuits involve capacitors charging or discharging through resistors. When you use a flash camera, it takes a few seconds to charge the capacitor that powers the flash. The light flash discharges the capacitor in a tiny fraction of a second. Why does charging take longer than discharging?

10 63 Charge In the circuit of Fig. 8.7, when the switch is in position 1, the capacitor of capacitance C charges through the resistor of resistance R from a 9V battery. The microammeter records the charging current I and the voltmeter reads the PD V c across the capacitor at different times t. Figure 8.7: Position 1: Charging, Position 2: Discharging. If readings of I and V c are taken at 10 second intervals and graphs of these quantities plotted against t, Fig. 8.8a and b, they show that Figure 8.8: (a) I vs t, (b) V vs t. 1. I has its maximum value at the start when the capacitor begins to charge and then decreases more and more slowly until it becomes zero.

11 64 CHAPTER 8. CAPACITORS 2. V c rises rapidly from zero and slowly approaches its maximum value (9V) which it reaches when the capacitor is fully charged and I = 0. If the readings of the PD V R across the resistor were also taken, a graph of V R against t would have the same shape as that of I against t (since V R = IR at all times) and is shown as the dashed graph in Fig. 8.8b. All three graphs are exponential curves; those of I and V R are decay curves and that of V c is a growth curve. A graph of the charge on the capacitor Q against t has the same shape as the V c against t graph since Q = V c C. V c = V 0 (1 e t/rc) (8.17) Q c = Q 0 (1 e t/rc) (8.18) I = I 0 (e t/rc) (8.19) Also note that during charging the sum of the voltages across the resistor and capacitor equals the battery voltage V, that is V = V c +V R. Initially V C = 0, so V = V R. Finally, when the capacitor is fully charged, I = 0, therefore V R = 0 and V C = V. Time constant If the charging current I remained steady at its starting value, a capacitor would be fully charged after time T = C R seconds which, for the circuit of Fig. 8.7, equals = 50s. In fact I decreases with time, as Fig. 8.8a shows, and the capacitor has only 0.63 of its full charge and PD after 50s. Nevertheless, T = CR, called the time constant, is a useful measure of how long it takes a capacitor to charge through a resistor. The greater the values of C and R, the greater is T and the more slowly the PD across it rises. In general, T = CR is the time for V C (and Q) to rise to 63% of the charging PD at the start of the time, V.

12 65 Discharge In Fig. 8.7, when the switch is moved from position 1 to position 2, the capacitor discharges through the resistor. If the graphs of I, V C and V R are plotted as before, they are again exponential curves, as shown in Figs 8.9a and b. Note that: 1. The discharge current I, and so also V R, are in the opposite direction to that during charge. 2. V C and V R are in opposition during discharge. Figure 8.9: (a) I vs t, (b) V vs t. Using calculus it can be shown that V C decays according to the equation V C = V 0 e t/cr (8.20) where V 0 is the PD across the capacitor initially and e, the base of natural logarithms, equal 2.7. Substituting t = T = CR we get V C = V 0 e = V = 0.37V 0 (8.21)

13 66 CHAPTER 8. CAPACITORS In this case the time constant T is the time for V C to fall to 37% of its value at the start of the discharge. Also, Q C = Q 0 e t/cr (8.22) CR is known as the time constant of the circuit and is the time taken for the charge on the capacitor to fall to 1/e (i.e. 36.8%) of its initial value. Since V C = V R, i.e. the PD across the capacitor is equal and opposite to that across the resistor. V R = V 0 e t/cr (8.23) I = (V 0 /R)e t/cr = (Q 0 /CR)e t/cr (8.24) i.e. the current during discharge decreases exponentially. When doing calculations the minus sign in Equ can be ignored; it merely means that the capacitor s charge Q is decreasing. Figure 8.10: This stop-motion photograph of a hummingbird feeding on a flower was obtained with an extremely brief and intense flash of light powered by the discharge of a capacitor through a gas. Now we can explain why the flash camera in our scenario takes so much longer to charge than discharge; the resistance while charging is significantly greater than while discharging. The internal resistance of the battery accounts for most of the resistance while charging. As the battery ages, the increasing internal resistance makes the charging process even slower.

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