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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: The terms inside the parentheses in the last two equations must be equal.These energies must be equal, because there is no other source and no other destination for energy in the circuit. This logic is valid in general for any number of resistors in series; thus, the total resistance of a series connection is Suppose the voltage output of the battery in [link] is , and the resistances are , , and . (d) Calculate the power dissipated by each resistor.
Thus the energy supplied by the source is , while that dissipated by the resistors is 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.
These two laws are directly involved in all electrical phenomena and will be invoked repeatedly to explain both specific effects and the general behavior of electricity. (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 This implies that the total or equivalent series resistance of three resistors is . (c) Calculate the voltage drop in each resistor, and show these add to equal the voltage output of the source.
Entering known values gives The total resistance with the correct number of significant digits is Discussion for (a) is, as predicted, less than the smallest individual resistance.
Strategy and Solution for (b) The total current can be found from Ohm’s law, substituting for the total resistance.
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 [link].Let the voltage output of the battery and resistances in the parallel connection in [link] be the same as the previously considered series connection: , , , and . (d) Calculate the power dissipated by each resistor.Strategy and Solution for (a) The total resistance for a parallel combination of resistors is found using the equation below.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 simplest combinations of resistors are the series and parallel connections illustrated in [link].The total resistance of a combination of resistors depends on both their individual values and how they are connected. For example, if current flows through a person holding a screwdriver and into the Earth, then in [link](a) could be the resistance of the screwdriver’s shaft, the resistance of its handle, the person’s body resistance, and the resistance of her shoes.[link] shows resistors in series connected to a source.It seems reasonable that the total resistance is the sum of the individual resistances, considering that the current has to pass through each resistor in sequence.Choosing , and entering the total current, yields More complex connections of resistors are sometimes just combinations of series and parallel.These are commonly encountered, especially when wire resistance is considered.