21.1 Resistors in series and parallel  (Page 3/18)

Strategy and Solution for (b)

The current is found using Ohm’s law, $V=\text{IR}$ . Entering the value of the applied voltage and the total resistance yields the current for the circuit:

Strategy and Solution for (c)

The voltage—or $\text{IR}$ drop—in a resistor is given by Ohm’s law. Entering the current and the value of the first resistance yields

Similarly,

and

Discussion for (c)

The three $\text{IR}$ drops add to $\text{12}\text{.}0\text{V}$ , as predicted:

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    , $P=\text{IV}$ , where $P$ is electric power. In this case, each resistor has the same full current flowing through it. By substituting Ohm’s law $V=\text{IR}$ into Joule’s law, we get the power dissipated by the first resistor as

Similarly,

and

Discussion for (d)

Power can also be calculated using either $P=\text{IV}$ or $P=\frac{V^2}{R}$ , where $V$ 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 $P=\text{IV}$ , where $V$ is the source voltage. This gives

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,

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.

Resistors in parallel

[link] shows resistors in parallel    , wired to a voltage source. Resistors are in parallel when each resistor is 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 source is not overloaded). For example, an automobile’s headlights, radio, and so on, are wired in parallel, so that they utilize the full voltage of the source and can operate completely independently. The same is true in your house, or any building. (See [link] (b).)

To find an expression for the equivalent parallel resistance $R_{\text{p}}$ , 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 $I_1=\frac{V}{R_1}$ , $I_2=\frac{V}{R_2}$ , and $I_3=\frac{V}{R_3}$ . Conservation of charge implies that the total current $I$ produced by the source is the sum of these currents: