# Electrical Circuits

## Electric Power


### Section Key Terms

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 ). 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 , 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  
   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,  
 , where  
   Q is the charge in coulombs and  
   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 . 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 , we used the resistance and Ohm’s law to find the current. Ohm’s law gives the current: 
 , 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  
  
 and use this to eliminate V in the equation  
 , we obtain
This gives the power in terms of only the current and the resistance.
Thus, by combining Ohm’s law with the equation  
   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 , 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  
  .

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### Section Summary

1. Electric power is dissipated in the resistances of a circuit. Capacitors do not dissipate electric power.
2. Electric power is proportional to the voltage and the current in a circuit.
3. Ohm’s law provides two extra expressions for electric power: one that does not involve current and one that does not involve voltage.

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