OCR A Physics A-level Topic 6.1: Capacitors PDF

OCR A Physics A-level Topic 6.1: Capacitors (Content in italics is not mentioned specifically in the course specification but is nevertheless topical, relevant and possibly examinable) www.pmt.education Definition of a Capaci...

OCR A Physics A-level Topic 6.1: Capacitors (Content in italics is not mentioned specifically in the course specification but is nevertheless topical, relevant and possibly examinable) www.pmt.education Definition of a Capacitor and Capacitance A capacitor is an electrical component that stores charge on two separated metallic plates. An insulator, sometimes called a dielectric, is placed between the plates to prevent the charge from travelling across the gap. The capacitance, 𝐶𝐶, is defined as the charge stored, 𝑄𝑄, per unit potential difference, 𝑉𝑉, across the two plates. Therefore we can write 𝑄𝑄 𝐶𝐶 = 𝑉𝑉 where capacitance is measured in Farads, F (CV-1). When a capacitor is connected to a DC power supply, e.g. a cell or battery, there is a brief current as the power supply draws electrons from one plate and deposits them on the other plate. This leaves the first plate with charge +𝑄𝑄 and the second with charge – 𝑄𝑄. These charges will be equal and opposite due to the conservation of charge. Current will flow in the circuit until the potential difference between the plates is equal to that of the electromotive force or e.m.f. of the power supply. Dielectric Insulators The dielectric has another purpose: to increase the capacitance of the device by polarizing in the electric field and effectively increasing the charge stored on the plates. Dielectrics have an associated electrical permittivity (see 6.2 Electric Fields) which describes its ability to polarize and strengthen the charge storage capability of the device. This is why in reality the insulator is rarely a vacuum or just air as these materials do not polarize well (or at all in the case of the vacuum) and so are poor dielectrics. www.pmt.education Capacitors in Series Kirchhoff’s voltage law states that the sum of the e.m.f.s in any closed loop in a circuit is equal to the sum of the potential differences in the same loop (see section 4.3). 𝑉𝑉 = 𝑉𝑉1 + 𝑉𝑉2 + 𝑉𝑉3 + ⋯ + 𝑉𝑉𝑁𝑁 𝑄𝑄 𝑄𝑄 From the equation 𝐶𝐶 = 𝑉𝑉 , it is clear that 𝑉𝑉 = 𝐶𝐶 and so substituting this into the expression for Kirchhoff’s voltage law gives 𝑄𝑄 𝑄𝑄 𝑄𝑄 𝑄𝑄 𝑄𝑄 = + + + ⋯+ 𝐶𝐶𝑇𝑇 𝐶𝐶1 𝐶𝐶2 𝐶𝐶3 𝐶𝐶𝑁𝑁 where 𝐶𝐶𝑇𝑇 is the combined capacitance of all the series capacitors. As 𝑄𝑄 is a constant it can be factorised out to give 1 1 1 1 1 = + + + ⋯+ 𝐶𝐶𝑇𝑇 𝐶𝐶1 𝐶𝐶2 𝐶𝐶3 𝐶𝐶𝑁𝑁 Therefore 1 1 1 1 −1 𝐶𝐶𝑇𝑇 = + + + ⋯+ 𝐶𝐶1 𝐶𝐶2 𝐶𝐶3 𝐶𝐶𝑁𝑁 Note that this equation is similar to the equation for the total resistance of a number of resistors in parallel. Capacitors in Parallel Kirchhoff’s current law states that the total current flowing into a node in a circuit must be equal to the total current flowing out of that node. Therefore, we can state that 𝐼𝐼𝑇𝑇 = 𝐼𝐼1 + 𝐼𝐼2 + 𝐼𝐼3 + ⋯ + 𝐼𝐼𝑁𝑁 Charge can be stated as 𝑄𝑄 = 𝐼𝐼𝐼𝐼, so using the above and factorising out the constant time, 𝑄𝑄𝑇𝑇 = 𝑄𝑄1 + 𝑄𝑄2 + 𝑄𝑄3 + ⋯ + 𝑄𝑄𝑁𝑁 www.pmt.education  𝑄𝑄 Finally, substituting the equation 𝐶𝐶 = 𝑉𝑉 and that the voltage is the same over each component in parallel we can write 𝐶𝐶𝑇𝑇 = 𝐶𝐶1 + 𝐶𝐶2 + 𝐶𝐶3 + ⋯ + 𝐶𝐶𝑁𝑁 Note that this equation is similar to the equation for the total resistance of a number of resistors in series. Energy Stored in a Capacitor Work must be done by the power supply to deposit negatively charged electrons onto the negative plate as like charges repel according to Coulomb’s law (see 6.2 Electric Fields). Equally, work is done to remove electrons from the positive plate as negative charges are attracted to positive regions. The graph below shows the charge stored on a capacitor plates against the potential difference over the device. As voltage is defined as the electrical potential energy per unit charge (see 6.2 Electric Fields), the area under the graph must therefore represent the work done in charging up the capacitor and so the energy stored in the capacitor. Therefore 1 𝑊𝑊 = 2 𝑄𝑄𝑄𝑄 𝑄𝑄 however, 𝑄𝑄 = 𝐶𝐶𝐶𝐶 and also 𝑉𝑉 = 𝐶𝐶 thus 1 𝑄𝑄 2 𝑊𝑊 = 2 𝑉𝑉 2 𝐶𝐶 𝑊𝑊 = 2𝐶𝐶 www.pmt.education Applications of Capacitors Capacitors are used to store and discharge large quantities of energy in a short time period. This makes them useful for short pulses of energy such as camera flashes and touch screens where a short finger press leads to a large buildup of energy in a capacitor. They are also integral to uninterrupted power supplies or UPSs which are used as backup power supplies when the mains electricity supply fails. UPSs are commonly found in data centers to protect the hardware and in hospitals to maintain a constant power supply to life support machines. Finally, capacitors are used in the process of converting alternating current (AC) into direct current (DC). Once a sinusoidal AC signal has passed through a full wave rectifier, the current flows in one direction but varies as shown. The current can then be passed through a smoothing circuit in which a capacitor stores energy as the p.d. rises and discharges as it falls. This can be used maintain a more constant current. The signal can then be passed through another smoothing circuit and another until the voltage is effectively constant. Charging and Discharging Capacitors Once a capacitor has been charged, it can then be discharged by disconnecting the power supply and connecting up another electrical component. This can be achieved by flipping the switch from in the circuit diagram so from A to B. Often, this component is a resistor as then the resistance, and so the time constant for the fall in voltage, can be known to a high degree of accuracy. When the power supply is disconnected, the electrons packed onto the negative plate are no longer subject to the e.m.f. which held them in such close proximity. They repel one another and so flow round circuit dissipating electric energy as heat in the resistor. Once, the charges on the negative and positive plates have equilibrated, there is no longer any potential difference across the capacitor (𝑄𝑄 = 0) and the electrons cease to flow resulting in the current dropping to zero. Naturally, this discharging process takes time. The time constant over which this discharging process occurs depends firstly on the capacitance and also on the magnitude of the resistance in the discharging circuit. The lower the resistance in the discharging circuit, the higher the current can be as current is indirectly proportional to www.pmt.education  1 the resistance from Ohm’s law (𝐼𝐼 ∝ 𝑅𝑅). If the current is higher, then the charge on the plates will fall to zero in a faster time as ∆𝑄𝑄 = 𝐼𝐼𝐼𝐼. Equally, the larger the capacitance the larger the charge stored per unit potential difference. As potential difference is proportional to the current by Ohm’s law, then capacitance is a measure of the charge stored per rate that charge flows from the plates that is to say 𝑄𝑄 𝑄𝑄 𝐶𝐶 = ∝ ~𝜏𝜏 𝑉𝑉 𝐼𝐼 where 𝜏𝜏 is the time period over which the capacitor discharges and the symbol ~ here means ‘goes as’ so not necessarily directly proportional but as one quantity increases so does the other. We use this symbol as the current is not constant over the time spent discharging so the relation is not as simple as 𝑄𝑄 = 𝐼𝐼𝐼𝐼. Simply put, this means that the amount of charge that can flow before the voltage drops to zero is higher and so a longer time is needed for the discharge to take place. Before the resistor is connected, the potential difference, 𝑉𝑉0, across the plates is at its 𝑄𝑄 maximum and given by 𝑉𝑉0 = 𝐶𝐶0 where 𝑄𝑄0 is the initial charge stored on the plates. At time 𝑡𝑡 = 0, the resistor circuit is connected and the current flowing through the circuit will be 𝑉𝑉0 /𝑅𝑅 as given by Ohm’s law. As the electrons flow, the charge stored will decrease as the negative plate loses electrons and the positive plate gains electrons. This in turn will decrease the potential difference over the capacitor and so current must also decrease and will eventually reach zero. Derivation of the relationship between Charge and Time in a Discharging Capacitor Current can be defined as the differential of charge with respect to time 𝑑𝑑𝑑𝑑 𝐼𝐼 = − 𝑑𝑑𝑑𝑑 where the negative sign is a result of conventional current being in the opposite direction to electron flow. Though as 𝑉𝑉 = 𝐼𝐼𝐼𝐼 and 𝑄𝑄 = 𝐶𝐶𝐶𝐶, www.pmt.education  𝑑𝑑𝑑𝑑 𝑄𝑄 =− 𝑑𝑑𝑑𝑑 𝐶𝐶𝐶𝐶 Separating variables and integrating from 𝑡𝑡 = 0 when 𝑄𝑄(𝑡𝑡 = 0) = 𝑄𝑄0 gives 𝑄𝑄 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = − 𝑄𝑄0 𝑄𝑄 0 𝐶𝐶𝐶𝐶 𝑄𝑄 𝑡𝑡 ln = − 𝑄𝑄0 𝐶𝐶𝐶𝐶 Therefore, taking the exponent 𝑡𝑡 𝑄𝑄 = 𝑄𝑄0 𝑒𝑒 −𝐶𝐶𝐶𝐶 Similar equations can be written for voltage and current as they are related to the 𝑑𝑑𝑑𝑑 charge by 𝐼𝐼 = 𝑑𝑑𝑑𝑑 and 𝑉𝑉 = 𝐼𝐼𝐼𝐼. 𝑡𝑡 𝑡𝑡 𝑉𝑉 = 𝑉𝑉0 𝑒𝑒 −𝐶𝐶𝐶𝐶 𝐼𝐼 = 𝐼𝐼0 𝑒𝑒 −𝐶𝐶𝐶𝐶 The relationship between 𝑉𝑉, 𝐼𝐼 or 𝑄𝑄 and 𝑡𝑡 is an exponential decay as seen in the graph on the right below. While charging a capacitor, at any time in the circuit e.m.f. (𝑉𝑉0) will be equal to the sum of the p.d.s across the resistor (𝑉𝑉𝑅𝑅 ) and the capacitor (𝑉𝑉𝐶𝐶 ) by Kirchhoff’s voltage law. 𝑉𝑉0 = 𝑉𝑉𝑅𝑅 + 𝑉𝑉𝐶𝐶 𝑡𝑡 by substituting in 𝑉𝑉𝑅𝑅 = 𝐼𝐼𝐼𝐼, and 𝐼𝐼 = 𝐼𝐼0 𝑒𝑒 −𝐶𝐶𝐶𝐶 𝑡𝑡 𝑉𝑉𝐶𝐶 = 𝑉𝑉0 − 𝐼𝐼0 𝑅𝑅𝑅𝑅 −𝐶𝐶𝐶𝐶 or 𝑡𝑡 𝑉𝑉𝐶𝐶 = 𝑉𝑉0 (1 − 𝑒𝑒 −𝐶𝐶𝐶𝐶 ) A similar expression can be written for the charge on the capacitor. This relationship is shown in the graph on the left below. www.pmt.education The value of the time constant of the circuit is seen in the exponents of the equations above and is often give the symbol 𝜏𝜏 = 𝐶𝐶𝐶𝐶. For a discharging capacitor, when 𝑡𝑡 = 𝜏𝜏 the charge on the capacitor will have decreased to approximately 37% of its original value. Graphing Variables in Capacitor-Resistor Circuits The change in charge with time can be graphed iteratively for a capacitor-resistor circuit. First, the time constant is calculated from the known values of the capacitance and the resistance. Then, using the equation ∆𝑄𝑄 𝑄𝑄 =− ∆𝑡𝑡 𝐶𝐶𝐶𝐶 it can be seen that, in a small time interval ∆𝑡𝑡 compared with 𝜏𝜏, the change in charge stored, ∆Q, can be calculated. From this a new charge stored can be calculated at the new time. 𝑄𝑄 𝑡𝑡𝑖𝑖+1 = 𝑡𝑡𝑖𝑖 + ∆𝑡𝑡 ∆𝑄𝑄𝑖𝑖+1 = − 𝐶𝐶𝐶𝐶𝑖𝑖 ∆𝑡𝑡 𝑄𝑄(𝑡𝑡𝑖𝑖+1 ) = 𝑄𝑄(𝑡𝑡𝑖𝑖 ) + ∆𝑄𝑄𝑖𝑖 This can be repeated for each new charge to give a value for the charge at each moment in time. These could be graphed to give the approximate behaviour of the charge stored with time. These exponential graphs also show that if 𝑉𝑉, 𝐼𝐼 or 𝑄𝑄 are measured at set time intervals that 𝑉𝑉1 𝑉𝑉 𝑉𝑉 = 𝑉𝑉2 = 𝑉𝑉3 … 𝑉𝑉0 1 2 or more generally 𝑉𝑉𝑖𝑖+1 𝑉𝑉𝑖𝑖+2 = 𝑉𝑉𝑖𝑖 𝑉𝑉𝑖𝑖+1 This relationship is always true for exponential decays. www.pmt.education Experimental Techniques to investigate Capacitor-Resistor Circuits To investigate the charge or discharge of a capacitor a circuit with a DC power supply, a capacitor, a resistor in series, an ammeter in series and a voltmeter in parallel are needed. Data loggers can be used to collect the data in time as capacitors often discharge very quickly. Plotting current and voltage with time in charging and discharging circuits can be used to investigate of the exponential relationships between the variables current and p.d. with time. The readings for the voltage and current should be taken at set intervals which should be small compared to the time constant. This can then allow for an experimental determination of the time constant. The experimental value could then be compared to the theoretical value based on the values of the resistance and capacitance. Dependence of Capacitance on Dimensions of the Capacitor The capacitance of a parallel plate capacitor depends on the number of electrons that can be stored on the negative plate and so is directly proportional to the area of the plates, 𝑨𝑨. The attraction between charges on the negative plate and the positive plate depends on the separation of the plates, 𝑑𝑑. Therefore the capacitance is indirectly proportional to 𝒅𝒅 so 1 𝐶𝐶 ∝ 𝐴𝐴 𝐶𝐶 ∝ 𝑑𝑑 With a vacuum between the two plates capacitance is then defined as 𝜀𝜀0 𝐴𝐴 𝐶𝐶 = 𝑑𝑑 For non-vacuum insulators this permittivity changes, such that 𝜀𝜀 = 𝜀𝜀𝑟𝑟 𝜀𝜀0 where 𝜀𝜀𝑟𝑟 is the relative permittivity of the dielectric medium. Hence, for a general parallel plate capacitor 𝜀𝜀𝜀𝜀 𝐶𝐶 = 𝑑𝑑 www.pmt.education