Power in Electric Circuits

Power is associated by many people with electricity. Knowing that power is the rate of energy use or energy conversion, what is the expression for electric power? Power transmission lines might come to mind. We also think of lightbulbs in terms of their power ratings in watts. Let us compare a 25-W bulb with a 60-W bulb. (See Figure 3.17(a)) Since both operate on the same voltage, the 60-W bulb must draw more current to have a greater power rating. Thus the 60-W bulb's resistance must be lower than that of a 25-W bulb. If we increase voltage, we also increase power. For example, when a 25-W bulb that is designed to operate on 120 V is connected to 240 V, it briefly glows very brightly and then burns out. Precisely how are voltage, current, and resistance related to electric power?

Figure 3.17 (a) Which of these lightbulbs, the 25-W bulb (upper left) or the 60-W bulb (upper right), has the higher resistance? Which draws more current? Which uses the most energy? Can you tell from the color that the 25-W filament is cooler? Is the brighter bulb a different color and if so why? (Dickbauch, Wikimedia Commons; Greg Westfall, Flickr) (b) This compact fluorescent light (CFL) puts out the same intensity of light as the 60-W bulb, but at 1/4 to 1/10 the input power. (dbgg1979, Flickr)

Electric energy depends on both the voltage involved and the charge moved. This is expressed most simply as $\text{PE}=\text{qV}\text{,}$ where $q$ is the charge moved and $V$ is the voltage (or more precisely, the potential difference the charge moves through). Power is the rate at which energy is moved, and so electric power is

Recognizing that current is $I=q/t$ (note that $\mathrm{\Delta }t=t$ here), the expression for power becomes

Electric power ($P\text{)}$ is simply the product of current times voltage. Power has familiar units of watts. Since the SI unit for potential energy (PE) is the joule, power has units of joules per second, or watts. Thus, $1A\cdot V=1\mathrm{W.}$ For example, cars often have one or more auxiliary power outlets with which you can charge a cell phone or other electronic devices. These outlets may be rated at 20 A, so that the circuit can deliver a maximum power $P=\text{IV}=(\text{20 A})(\text{12 V})=\text{240 W.}$ In some applications, electric power may be expressed as volt-amperes or even kilovolt-amperes $\text{(}\text{1 kA}\cdot \text{V}=\text{1 kW}\text{).}$

To see the relationship of power to resistance, we combine Ohm's law with $P=\text{IV.}$ Substituting $I=\text{V/R}$ gives $P=(V/R)V=V^2\text{/}R\text{.}$ Similarly, substituting $V=\text{IR}$ gives $P=I(\text{IR})=I^{2}R\text{.}$ Three expressions for electric power are listed together here for convenience.

Note that the first equation is always valid, whereas the other two can be used only for resistors. In a simple circuit, with one voltage source and a single resistor, the power supplied by the voltage source and that dissipated by the resistor are identical. In more complicated circuits, $P$ can be the power dissipated by a single device and not the total power in the circuit.

Making Connections: Using Graphs to Calculate Resistance

As $p\propto I^2$ and $p\propto V^2$, the graph for power versus current or voltage is quadratic. An example is shown in the figure below.

Figure 3.18 The figure shows (a) power versus current and (b) power versus voltage relationships for simple resistor circuits.

Using equations (20.29) and (20.30), we can calculate the resistance in each case. In graph (a), the power is 50 W when current is 5 A; hence, the resistance can be calculated as $R=P\text{/}{I}^2=50\text{/}{5}^2=2\text{ Ω.}$ Similarly, the resistance value can be calculated in graph (b) as $R=V^2\text{/}P={10}^2\text{/}50=2\text{ Ω}$

Different insights can be gained from the three different expressions for electric power. For example, $P=V^2/R$ implies that the lower the resistance connected to a given voltage source, the greater the power delivered. Furthermore, since voltage is squared in $P=V^2/R\text{,}$ the effect of applying a higher voltage is perhaps greater than expected. Thus, when the voltage is doubled to a 25-W bulb, its power nearly quadruples to about 100 W, burning it out. If the bulb's resistance remained constant, its power would be exactly 100 W, but at the higher temperature its resistance is higher, too.

The Cost of Electricity

The more electric appliances you use and the longer they are left on, the higher your electric bill. This familiar fact is based on the relationship between energy and power. You pay for the energy used. Since $P=E/t\text{,}$ we see that

is the energy used by a device using power $P$ for a time interval $\mathrm{t.}$ For example, the more lightbulbs burning, the greater $P$ used; the longer they are on, the greater $t$ is. The energy unit on electric bills is the kilowatt-hour ($\text{kW}\cdot \text{h}$), consistent with the relationship $E=\text{Pt.}$ It is easy to estimate the cost of operating electric appliances if you have some idea of their power consumption rate in watts or kilowatts, the time they are on in hours, and the cost per kilowatt-hour for your electric utility. Kilowatt-hours, like all other specialized energy units such as food calories, can be converted to joules. You can prove to yourself that $\text{1 kW}\cdot \text{h = 3}\text{.}6\times {\text{10}}^6\text{J.}$

The electrical energy $\text{(}E\text{)}$) used can be reduced either by reducing the time of use or by reducing the power consumption of that appliance or fixture. This will not only reduce the cost, but it will also result in a reduced impact on the environment. Improvements to lighting are some of the fastest ways to reduce the electrical energy used in a home or business. About 20 percent of a home's use of energy goes to lighting, while the number for commercial establishments is closer to 40 percent. Fluorescent lights are about four times more efficient than incandescent lights—this is true for both the long tubes and the compact fluorescent lights (CFL). (See Figure 3.17(b)) Thus, a 60-W incandescent bulb can be replaced by a 15-W CFL, which has the same brightness and color. CFLs have a bent tube inside a globe or a spiral-shaped tube, all connected to a standard screw-in base that fits standard incandescent light sockets. (Original problems with color, flicker, shape, and high initial investment for CFLs have been addressed in recent years.) The heat transfer from these CFLs is less, and they last up to 10 times longer. The significance of an investment in such bulbs is addressed in the next example. New white LED lights (which are clusters of small LED bulbs) are even more efficient (twice that of CFLs) and last 5 times longer than CFLs. However, their cost is still high.