Resistors in Series and Parallel Quiz

PDF page 990: These energies must be equal, because there is no other source and no other destination for energy in the circuit. Thus, . The charge cancels, yielding , as stated. (Note that the same amount of charge passes through the battery and each resistor in a given amount of time, since there is no capacitance to store charge, there is no place for charge to leak, and charge is conserved.) Now substituting the values for the individual voltages gives 21.3 Note that for the equivalent single series resistance , we have 21.4 This implies that the total or equivalent series resistance of three resistors is . This logic is valid in general for any number of resistors in series; thus, the total resistance of a series connection is 21.5 as proposed. Since all of the current must pass through each resistor, it experiences the resistance of each, and resistances in series simply add up.

PDF page 989: Figure 21.3 shows resistors in series connected to a voltage source. It seems reasonable that the total resistance is

PDF page 989: To verify that resistances in series do indeed add, let us consider the loss of electrical power, called a voltage drop, in each resistor in Figure 21.3. According to Ohm’s law, the voltage drop, , across a resistor when a current flows through it is calculated using the equation , where equals the current in amps (A) and is the resistance in ohms . Another way to think of this is that is the voltage necessary to make a current flow through a resistance . So the voltage drop across is , that across is , and that across is . The sum of these voltages equals the voltage output of the source; that is, 21.1 This equation is based on the conservation of energy and conservation of charge. Electrical potential energy can be described by the equation , where is the electric charge and is the voltage. Thus the energy supplied by the source is , while that dissipated by the resistors is 21.2

PDF page 989: The derivations of the expressions for series and parallel resistance are based on the laws of conservation of energy and conservation of charge, which state that total charge and total energy are constant in any process.

PDF page 991: connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it. Each resistor draws the same current it would if it alone were connected to the voltage source (provided the voltage

PDF page 993: This relationship results in a total resistance that is less than the smallest of the individual resistances. (This is seen in the next example.) When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, and so the total resistance is lower.

PDF page 993: This relationship results in a total resistance that is less than the smallest of the individual resistances. (This is seen in the next example.) When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, and so the total resistance is lower.

PDF page 991: source is not overloaded). For example, an automobile’s headlights, radio, and so on, are wired in parallel, so that

PDF page 991: they utilize the full voltage of the source and can operate completely independently. The same is true in your house,

PDF page 990: These energies must be equal, because there is no other source and no other destination for energy in the circuit. Thus, . The charge cancels, yielding , as stated. (Note that the same amount of charge passes through the battery and each resistor in a given amount of time, since there is no capacitance to store charge, there is no place for charge to leak, and charge is conserved.) Now substituting the values for the individual voltages gives 21.3 Note that for the equivalent single series resistance , we have 21.4 This implies that the total or equivalent series resistance of three resistors is . This logic is valid in general for any number of resistors in series; thus, the total resistance of a series connection is 21.5 as proposed. Since all of the current must pass through each resistor, it experiences the resistance of each, and resistances in series simply add up.

PDF page 992: setup in a house. (credit: Dmitry G, Wikimedia Commons) To find an expression for the equivalent parallel resistance , let us consider the currents that flow and how they are related to resistance. Since each resistor in the circuit has the full voltage, the currents flowing through the individual resistors are , , and . Conservation of charge implies that the total current produced by the source is the sum of these currents: 21.17 Substituting the expressions for the individual currents gives

PDF page 994: Discussion for (c) The total current is the sum of the individual currents: 21.28 This is consistent with conservation of charge. Strategy and Solution for (d) The power dissipated by each resistor can be found using any of the equations relating power to current, voltage, and resistance, since all three are known. Let us use , since each resistor gets full voltage. Thus,

PDF page 994: Discussion for (d) The power dissipated by each resistor is considerably higher in parallel than when connected in series to the same voltage source. Strategy and Solution for (e) The total power can also be calculated in several ways. Choosing , and entering the total current, yields 21.32 Discussion for (e) Total power dissipated by the resistors is also 179 W: 21.33 This is consistent with the law of conservation of energy. Overall Discussion Note that both the currents and powers in parallel connections are greater than for the same devices in series.

PDF page 994: , and it is smaller than any individual resistance in the combination.

PDF page 991: Power can also be calculated using either or , where is the voltage drop across the resistor (not the full voltage of the source). The same values will be obtained. Strategy and Solution for (e) The easiest way to calculate power output of the source is to use , where is the source voltage. This gives 21.15 Discussion for (e) Note, coincidentally, that the total power dissipated by the resistors is also 7.20 W, the same as the power put out by the source. That is, 21.16 Power is energy per unit time (watts), and so conservation of energy requires the power output of the source to be equal to the total power dissipated by the resistors.

PDF page 995: More complex connections of resistors are sometimes just combinations of series and parallel. These are commonly encountered, especially when wire resistance is considered. In that case, wire resistance is in series with other resistances that are in parallel. Combinations of series and parallel can be reduced to a single equivalent resistance using the technique illustrated in Figure 21.5. Various parts are identified as either series or parallel, reduced to their equivalents, and further reduced until a single resistance is left. The process is more time consuming than difficult.