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CHAPTER 20 - ELECTRIC CIRCUITS Electromotive force(emf) of a battery is the maximum voltage that can be produced by the battery. It is measured with no current flowing in the circuit. A car battery has an emf of 12.0 v and a flashlight battery has an emf of 1.5 v. Flashlight battery is really a misnomer since it consists internally of only one pair of electrodes. The car battery has 6 sets of electrodes(plates) arranged in series. Electric current is defined as the rate at which charge flows past a point in a conductor. The equation for average or constant current is: I = Δq/Δt where I is current, Δq is the amount of charge, and Δt is the time for that charge to pass by. The unit of electric current is the ampere(A) which is one coulomb per second. A defibrillator is used during a heart attack and passes 18.0 amps of current through a patient in 2.0 ms. How much charge passes through the patient? If the defibrillator voltage is 2500v, how much energy is delivered to the patient? Two types of electric current are direct current(dc) and alternating current(ac). Dc is produced by batteries and the current always flows in the same direction. In the case of ac, the current changes direction(+ to -) periodically. In the U.S. we use 60 Hz ac. Today we know that electric current is carried by the electrons that move in conductors. Early pioneers in the study of electricity, however, thought that electric current was carried by positive charge. Conventional current describes the flow of electricity from the positive terminal to the negative terminal of a source of current. Using this idea does not affect the amount or rate of energy transfer. We still use the same equations. Note in this text, I stands for conventional current. Ohm's Law Ohm's Law relates current, voltage and resistance of a circuit. It states that the current in a circuit is directly proportional to the voltage across the circuit and inversely proportional to the resistance in the circuit. The equation is: I = V/R V= IR The equation may be seen in either form. Resistance is defined as the tendency of a circuit component to oppose the flow of electric current. The SI unit of resistance is the Ohm(Ω) which is one volt per ampere. A device that provides resistance in an electric circuit is called a resistor. Above, we see a drawing of a flashlight and its bulb along with the circuit diagram. Example The filament of a light bulb has a resistance of 580 ohms. A voltage of 120 v is connected across the filament. How much current flows in the filament? The resistance of a circuit component depends on several factors. For most materials the resistance is directly proportional to the length, inversely proportional to the crosssectional area, and directly proportional to the resistivity. Resistivity is a proportionality constant that is a characteristic of any conductor. It does vary with temperature. The equation relating these variables is: R = ρL/A where ρ is resistivity. The metric unit for resistivity is the ohmmeter. Example A cylindrical copper cable carries a current of 1200 A. There is a potential difference of .016 v between two points on the conductor that are 0.24 m apart. Find the radius of the cable. (ρ = 1.72 x 10-8 ohm m) Superconductors are materials whose resistivity goes to zero at very low temperatures called the critical temperature. For most common metals, the critical temperature is less than 10 Kelvins. Some copper oxide complexes have a critical temperature as high as 175 K. Electric Power Electric power, just like mechanical power, is work per unit of time. Since the energy transferred depends on the voltage and the amount of charge, the equation can be written as: P = W/t = qV/t P = VI where V is voltage and I is current. Since according to Ohm's Law, V = IR, P = I2R P = V2/R Any of these equations can be used to calculate the power expended in a circuit component. Example An electric heater is used to heat small amounts of water and consists of a 15 ohm coil immersed directly in the water to be heated. it operates at 120 V. How long does it take to heat 0.50 kg of water from 13° C to 100° C? c = 4186 j/kgC° Alternating Current The second type of electric current is alternating current in which the direction of the current changes periodically. In the U.S., 60 complete cycles of change occur every second. This is the reason it is called 60 cycle or 60 Hz AC. In the diagram above, we can see how the voltage across the circuit varies from +V0 to -V0 in a sinusoidal fashion. In fact, we can write an equation relating the voltage and time using the sine function. V = V0sin2πft where V is the voltage as a function of time, V0 is the maximum magnitude of the voltage, f is the frequency in Hz and t is the time in seconds. Example For a frequency of 60 Hz, what is the smallest value of time when the voltage = one half of the peak value? In an AC circuit that has only resistance components, the current oscillates in phase with the voltage. The equation for current as a function of time in such a circuit is: I = I0sin2πft since I0 = V0/R. The power in an AC circuit is calculated as P = VI and varies with time as both current and voltage do. The equation is: P = I0V0sin22πft You can see that the power always has a positive value even though the voltage and current can have negative values. This happens because sin22πft(a squared term) is always positive. The diagram above shows that the average power is ½ of the maximum or peak power. A type of average voltage and average current can be derived from the equation for average power. Pave = (½)I0V0 Pave = I0 · V0 = Irms · Vrms √2 √2 where rms stands for root mean square. Irms and Vrms can be calculated from the maximum values for current and voltage. In the US, maximum AC voltage from a wall outlet is 170 volts which makes the rms voltage 120 volts. Example A light bulb is connected to a 120 V outlet. The equation for the current is I = (0.707 A)sin[(314 Hz)t]. (a)Find the frequency of the alternating current. (b)Determine the resistance of the bulb's filament. (c) Find the average power consumed by the light bulb. Series Wiring A series circuit is wired so that there is only one path for the electric current to follow. Each device in the circuit has the same current through it. In the circuit shown above, we can see how the current must flow through each of the light bulbs. Since charge cannot accumulate at any one of the light bulbs, it must continue to flow at the same rate through all three. The equivalent resistance for a series circuit is simply the sum of the individual component resistances. The equation is: Req = R1 + R2 + R3 This means that we could replace the three resistors in series with Req and the current in the circuit would not be changed. Example Three resistors, 25, 45, and 75 ohms are connected in series and a .51 amp current passes through them. Find the equivalent resistance and the potential difference across the three resistors. Parallel Wiring When devices are wired so that the same voltage is applied across each device, they are said to be in parallel. One disadvantage to series wiring is that all components in the circuit must work for current to flow in the circuit. With parallel wiring each branch forms a complete circuit independently of the others. In the diagram above, you can see that each light bulb forms a complete circuit whether the others are included or not. As in a series circuit, the resistors in a parallel circuit can be replaced with an equivalent resistance without changing the total voltage or current in the circuit. In a parallel circuit the total current equals the sum of the currents in each branch. It = I1 + I2 + I3 We can replace each current symbol with its corresponding V/R. Vt/Req = V1/R1 + V2/R2 + V3/R3 Since the total voltage and the voltages across each resistor are all the same, we can divide out all the voltages leaving: 1/Req = 1/R1 + 1/R2 + 1/R3 This equation can be used to find the equivalent resistance in a parallel circuit. Example A resistor(resistance = R) is connected first in parallel and then in series with a 2.00 ohm resistor. A battery delivers 5 times as much current to the parallel circuit as it does to the series circuit. Find the 2 possible values for R. Circuits Wired Partly in Series and Partly in Parallel When analyzing a circuit with both parallel and series components, determine the equivalent resistance for one part of the circuit at a time until you find the overall equivalent resistance. Example Find the equivalent resistance between points A and B in the diagram. Internal Resistance Any device in a circuit will have some resistance to the flow of electric current. Even devices that supply voltage to the circuit have resistance. In this case, it is called internal resistance. In many cases this resistance is too small to affect the output voltage of the device. However, the larger the current, the larger the voltage drop caused by the internal resistance. Terminal voltage is the voltage measured between two terminals of a battery. When no current is flowing, the terminal voltage is equal to the emf(maximum voltage) of the battery. The terminal voltage with current flowing is equal to the emf minus the product of the current and the internal resistance. Example A battery has an emf of 12.0 v and an internal resistance of 0.15 ohms. Find the terminal voltage when the battery is connected to a 1.50 ohm resistor. Kirchoff's Rules 1. Junction Rule - the total current entering a circuit junction must equal the total current leaving a circuit junction. In the example, 7 A flows into the junction so a total of 7 A must flow out. 2. Loop Rule - the sum of all the potential drops and rises around a closed loop in a circuit must add up to zero. This results from the Law of Conservation of Energy. Each potential drop can be calculated by multiplying the magnitude of the resistance by the magnitude of the current through it. To solve a problem using Kirchoff's rules follow these steps: 1. Draw the circuit diagram showing the current in each branch of the circuit. If you get the current direction wrong in any particular branch, your answer for that branch will be negative. Simply reverse the current direction. 2. Use conventional current which always flows from positive to negative. Mark both ends of each resistor + or - to help you see the current direction. 3. Apply the junction rule and the loop rule to the circuit to obtain as many independent equations as you have variables. 4. Solve the system of equations you generated in part 3. Example Determine the voltage across the 5.0 ohm resistor in the diagram. Which end of the resistor is at the higher potential? A galvanometer is a device that can be used to detect a voltage or an electric current. The device is wired with a large series resistance to be used as a voltmeter to measure voltage. It is wired with a very small parallel resistance(shunt) to be used as an ammeter to measure current. A voltmeter is always wired parallel to the device over which a voltage is to be measured. An ammeter is always wired in series with the part of the circuit in which current is to be measured. Capacitors in Series and Parallel Capacitors wired in parallel act like on larger capacitor. If they are wired in parallel the area of the plates is effectively increased therefore causing the combination to have a larger total capacitance than any individual capacitor in the circuit. The equation is: CT = C1 + C2 + C3 The total charge on these capacitors will also be equal to the sum of the individual charges while the voltage across each capacitor will be the same. In the case of capacitors arranged in series in a circuit, the charge on each capacitor is the same while the voltage drops are different and depend on the values of the individual capacitances. The equation for the equivalent capacitance when they are in series is: 1/CT = 1/C1 + 1/C2 + 1/C3 Example Determine the equivalent capacitance between A and B for the capacitors in the diagram. P 638 Questions 1, 2, 4, 5, 7, 9, 10, 12, 14, 15 P 639 Problems 1, 2, 4, 7, 8, 10, 11, 12, 13, 21, 24, 28, 30, 31, 35, 36, 39, 41, 43, 47, 48, 49, 51, 52, 54, 55, 58, 60, 63, 66, 67, 69, 73, 74, 77, 86, 87, 88, 110