Resistors in Series
When are resistors in series? Resistors are in series whenever the flow of charge, called the current, must flow through devices sequentially. For example, if current flows through a person holding a screwdriver and into the Earth, then in Figure 4.2(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.
Figure 4.3 shows resistors in series connected to a voltage 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. This fact would be an advantage to a person wishing to avoid an electrical shock, who could reduce the current by wearing high-resistance rubber-soled shoes. It could be a disadvantage if one of the resistances were a faulty high-resistance cord to an appliance that would reduce the operating current.
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 4.3.
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,
4.1
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
4.2
Connections: Conservation Laws
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.
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
4.3
Note that for the equivalent single series resistance we have
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
4.5
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.
Example 4.1 Calculating Resistance, Current, Voltage Drop, and Power Dissipation: Analysis of a Series Circuit
Suppose the voltage output of the battery in Figure 4.3 is and the resistances are and (a) What is the total resistance? (b) Find the current. (c) Calculate the voltage drop in each resistor and show that these add to equal the voltage output of the source. (d) Calculate the power dissipated by each resistor. (e) Find the power output of the source and show that it equals the total power dissipated by the resistors.
Strategy and Solution for (a)
The total resistance is simply the sum of the individual resistances, as given by the equation
4.6
Strategy and Solution for (b)
The current is found using Ohm’s law, Entering the value of the applied voltage and the total resistance yields the current for the circuit.
4.7
Strategy and Solution for (c)
The voltage—or drop—in a resistor is given by Ohm’s law. Entering the current and the value of the first resistance yields
4.8
Similarly,
4.9
and
4.10
Discussion for (c)
The three drops add to as predicted.
4.11
Strategy and Solution for (d)
The easiest way to calculate power in watts (W) dissipated by a resistor in a DC circuit is to use Joule’s law, where is electric power. In this case, each resistor has the same full current flowing through it. By substituting Ohm’s law into Joule’s law, we get the power dissipated by the first resistor as
4.12
Similarly,
4.13
and
4.14
Discussion for (d)
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
4.15
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,
4.16
Power is energy per unit time (watts), so conservation of energy requires the power output of the source to be equal to the total power dissipated by the resistors.
Major Features of Resistors in Series
- Series resistances add
- The same current flows through each resistor in series.
- Individual resistors in series do not get the total source voltage but rather divide it.