# Learning Objectives

### Learning Objectives

By the end of this section, you will be able to do the following:

- Define electric power and describe the electric power equation
- Calculate electric power in circuits of resistors in series, parallel, and complex arrangements

electric power |

Power is associated by many people with electricity. Every day, we use electric power to run our modern appliances. Electric power transmission lines are visible examples of electricity providing power. We also use electric power to start our cars, to run our computers, or to light our homes. Power is the rate at which energy of any type is transferred; electric power is the rate at which electric energy is transferred in a circuit. In this section, we’ll learn not only what this means, but also what factors determine electric power.

To get started, let’s think of light bulbs, which are often characterized in terms of their power ratings in watts. Let us compare a 25-W bulb with a 60-W bulb (see Figure 19.23). Although both operate at the same voltage, the 60-W bulb emits more light intensity than the 25-W bulb. This tells us that something other than voltage determines the power output of an electric circuit.

Incandescent light bulbs, such as the two shown in Figure 19.23, are essentially resistors that heat up when current flows through them and they get so hot that they emit visible and invisible light. Thus the two light bulbs in the photo can be considered as two different resistors. In a simple circuit such as a light bulb with a voltage applied to it, the resistance determines the current by Ohm’s law, so we can see that current as well as voltage must determine the power.

The formula for power may be found by dimensional analysis. Consider the units of power. In the SI system, power is given in watts (W), which is energy per unit time, or J/s

Recall now that a voltage is the potential energy per unit charge, which means that voltage has units of J/C

We can rewrite this equation as $\text{J}=\text{V}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}\text{C}$ and substitute this into the equation for watts to get

But a Coulomb per second (C/s) is an electric current, which we can see from the definition of electric current, $I=\frac{\Delta Q}{\Delta t}$, where $\Delta $ *Q* is the charge in coulombs and $\Delta $ *t* is time in seconds. Thus, equation above tells us that electric power is voltage times current, or

This equation gives the electric power consumed by a circuit with a voltage drop of *V* and a current of *I*.

For example, consider the circuit in Figure 19.24. From Ohm’s law, the current running through the circuit is

Thus, the power consumed by the circuit is

Where does this power go? In this circuit, the power goes primarily into heating the resistor in this circuit.

In calculating the power in the circuit of Figure 19.24, we used the resistance and Ohm’s law to find the current. Ohm’s law gives the current:$I=V/R$, which we can insert into the equation for electric power to obtain

This gives the power in terms of only the voltage and the resistance.

We can also use Ohm’s law to eliminate the voltage in the equation for electric power and obtain an expression for power in terms of just the current and the resistance. If we write Ohm’s law as $V=IR$
and use this to eliminate *V* in the equation $P=IV$, we obtain

This gives the power in terms of only the current and the resistance.

Thus, by combining Ohm’s law with the equation $P=IV$ for electric power, we obtain two more expressions for power: one in terms of voltage and resistance and one in terms of current and resistance. Note that only resistance (not capacitance or anything else), current, and voltage enter into the expressions for electric power. This means that the physical characteristic of a circuit that determines how much power it dissipates is its resistance. Any capacitors in the circuit do not dissipate electric power—on the contrary, capacitors either store electric energy or release electric energy back to the circuit.

To clarify how voltage, resistance, current, and power are all related, consider Figure 19.25, which shows the *formula wheel*. The quantities in the center quarter circle are equal to the quantities in the corresponding outer quarter circle. For example, to express a potential V in terms of power and current, we see from the formula wheel that $V=P/I$.

### Worked Example

#### Find the Resistance of a Lightbulb

A typical older incandescent lightbulb was 60 W. Assuming that 120 V is applied across the lightbulb, what is the current through the lightbulb?

### STRATEGY

We are given the voltage and the power output of a simple circuit containing a lightbulb, so we can use the equation $P=IV$ to find the current *I* that flows through the lightbulb.

Solving $P=IV$ for the current and inserting the given values for voltage and power gives

Thus, a half ampere flows through the lightbulb when 120 V is applied across it.

This is a significant current. Recall that household power is AC and not DC, so the 120 V supplied by household sockets is an alternating power, not a constant power. The 120 V is actually the time-averaged power provided by such sockets. Thus, the average current going through the light bulb over a period of time longer than a few seconds is 0.50 A.

### Worked Example

#### Boot Warmers

To warm your boots on cold days, you decide to sew a circuit with some resistors into the insole of your boots. You want 10 W of heat output from the resistors in each insole, and you want to run them from two 9-V batteries (connected in series). What total resistance should you put in each insole?

### STRATEGY

We know the desired power and the voltage (18 V, because we have two 9-V batteries connected in series), so we can use the equation $P={V}^{2}/R$ to find the requisite resistance.

Solving $P={V}^{2}/R$ for the resistance and inserting the given voltage and power, we obtain

Thus, the total resistance in each insole should be 32 $\text{\Omega .}$

Let’s see how much current would run through this circuit. We have 18 V applied across a resistance of 32 $\text{\Omega}$, so Ohm’s law gives

All batteries have labels that say how much charge they can deliver (in terms of a current multiplied by a time). A typical 9-V alkaline battery can deliver a charge of 565 $\text{mA}\cdot \text{h}$ (so two 9 V batteries deliver 1,130 $\text{mA}\cdot \text{h}$), so this heating system would function for a time of

### Worked Example

#### Power through a Branch of a Circuit

Each resistor in the circuit below is 30 $\text{\Omega}$. What power is dissipated by the middle branch of the circuit?

### STRATEGY

The middle branch of the circuit contains resistors ${R}_{\text{3}}\text{and}{R}_{5}$ in series. The voltage across this branch is 12 V. We will first find the equivalent resistance in this branch, and then use $P={V}^{2}/R$ to find the power dissipated in the branch.

The equivalent resistance is ${R}_{\text{middle}}={R}_{3}+{R}_{5}=30\text{\hspace{0.17em}}\text{\Omega}+30\text{\hspace{0.17em}}\text{\Omega}=60\text{\hspace{0.17em}}\text{\Omega}$. The power dissipated by the middle branch of the circuit is

Let’s see if energy is conserved in this circuit by comparing the power dissipated in the circuit to the power supplied by the battery. First, the equivalent resistance of the left branch is

The power through the left branch is

The right branch contains only ${R}_{6}$, so the equivalent resistance is ${R}_{\text{right}}={R}_{6}=30\text{\hspace{0.17em}}\text{\Omega}$. The power through the right branch is

The total power dissipated by the circuit is the sum of the powers dissipated in each branch.

The power provided by the battery is

where *I* is the total current flowing through the battery. We must therefore add up the currents going through each branch to obtain *I*. The branches contributes currents of

The total current is

and the power provided by the battery is

This is the same power as is dissipated in the resistors of the circuit, which shows that energy is conserved in this circuit.

# Practice Problems

### Practice Problems

What is the formula for the power dissipated in a resistor?

- The formula for the power dissipated in a resistor is $P=\frac{I}{V}\text{.}$
- The formula for the power dissipated in a resistor is $P=\frac{V}{I}\text{.}$
- The formula for the power dissipated in a resistor is
*P*=*IV*. - The formula for the power dissipated in a resistor is
*P*=*I*^{2}*V*.

What is the formula for power dissipated by a resistor given its resistance and the voltage across it?

- The formula for the power dissipated in a resistor is $P=\frac{R}{{V}^{2}}$
- The formula for the power dissipated in a resistor is $P={V}^{2}R$
- The formula for the power dissipated in a resistor is $P=\frac{{V}^{2}}{R}$
- The formula for the power dissipated in a resistor is $P={I}^{2}R$

# Check your Understanding

### Check your Understanding

Which circuit elements dissipate power?

- capacitors
- inductors
- ideal switches
- resistors

- Electric power is proportional to current through the resistor multiplied by the square of the voltage across the resistor.
- Electric power is proportional to square of current through the resistor multiplied by the voltage across the resistor.
- Electric power is proportional to current through the resistor divided by the voltage across the resistor.
- Electric power is proportional to current through the resistor multiplied by the voltage across the resistor.