Electric circuits are designed to serve a useful
function. The mere movement of charge from terminal to
terminal
is of little use if the electrical energy possessed by the
charge is not transformed into another useful form. To equip
a circuit with a battery and a wire leading from positive to
negative terminal without an electrical device (light bulb,
beeper, motor, etc.) would lead to a high rate of charge
flow. Such a circuit is referred to as a short
circuit. With charge flowing rapidly between terminals,
the rate at which energy would be consumed would be high.
Such a circuit would heat the wires to a high temperature
and drain the battery of its energy rather quickly. When a
circuit is equipped with a light bulb, beeper, or motor, the
electrical energy supplied to the charge by the battery is
transformed into other forms in the electrical device. A
light bulb, beeper and motor are generally referred to as a
load. In a light bulb,
electrical energy is transformed into useful light energy
(and some non-useful thermal energy). In a beeper,
electrical energy is transformed into sound energy. And in a
motor, electrical energy is transformed into mechanical
energy.

An electrical circuit is simply an energy
transformation tool. Energy is provided to the circuit by an
electrochemical cell, battery, generator or other electrical
energy source. And energy is delivered by the circuit to the
load at the location of the load. The rate at which this
energy transformation occurs is of great importance to those
who design electrical circuits for useful functions.
Power - the rate at which
mechanical work is done - was introduced in Unit
5 of the Physics Classroom. Here, we will discuss power
in electrical terms; while the context has changed, the
essential meaning of the concept of power will remain the
same. Power is the rate
at which electrical energy is supplied to a circuit or
consumed by a load. The electrical energy is supplied to the
load by an energy source such as an electrochemical cell.
Recall from Lesson 1 that
a cell does work upon a charge to move it from the low
energy to the high energy terminal. The work done on the
charge is equivalent to the electrical potential energy
change of the charge. Thus, electrical power, like
mechanical power, is the rate at which work is done. Like
current, power is a rate quantity. It's mathematical formula
is expressed on a per time basis.

Whether the focus is the energy gained by
the charge at the energy source or the energy lost by the
charge at the load, electrical power refers to the rate at
which the charge changes its energy.
In an electrochemical cell (or other energy source), the
change is a positive change (i.e., a gain in energy) and at
the load, the change is a negative change (i.e., a loss in
energy). Thus, power is often referred to as the rate of
energy change and its equation is expressed as the energy
change per time. Like mechanical power, the unit of
electrical power is the
watt, abbreviated
W. (Quite obviously, it
is important that the symbol
W as the unit of power
not be confused with the symbol
W for the quantity of
work done upon a charge by the energy source.) A watt of
power is equivalent to the delivery of 1 joule of energy
every second. In other words:

1 watt = 1 joule /
second

When it is observed that a light bulb is rated at 60
watts, then there are 60 joules of energy delivered to the
light bulb every second. A 120-watt light bulbs draws 120
joules of energy every second. The ratio of the energy
delivered or expended by the device to time is equal to the
wattage of the device.

The
kilowatt-hour

Electrical utility companies who provide energy for homes
provide a monthly bill charging those homes for the
electrical energy which they used. A typical bill can be
very complicated with a number of line items indicating
charges for various aspects of the utility service. But
somewhere on the bill will be a charge for the number of
kilowatt-hours of electricity which were consumed.
Exactly what is a kilowatt-hour? Is it a unit of power?
time? energy? or some other quantity? And when we pay for
the electricity which we use, what exactly is it that we are
paying for?

A careful inspection of the unit
kilowatt-hour reveals the answers to these questions.
A kilowatt is a unit of power and an hour is a unit of time.
So a kilowatt
• hour is a unit of Power • time. If Power =
Energy
/ time, then Power • time = Energy.
So a unit of power • time is a unit of energy. The
kilowatt • hour is a unit of energy. When an electrical
utility company charges a household for the electricity
which they used, they are charging them for electrical
energy. A utility company in the United States is
responsible for assuring that the electric potential
difference across the two main wires of the house is 110 to
120 volts. And maintaining this difference in potential
requires energy.

It is a common misconception that the
utility company provides electricity in the form of charge
carriers or electrons. The fact is that the mobile electrons
which are in the wires of our homes would be there whether
there was a utility company or not. The electrons come with
the atoms that make up the wires of our household circuits.
The utility company simply provides the energy which causes
the motion of the charge carriers within the household
circuits. And when they charge us for a few hundred
kilowatt-hours of electricity, they are providing us with an
energy bill.

Calculating
Power

The rate at which energy is delivered to a light bulb by
a circuit is related to the electric potential difference
established across the ends of the circuit (i.e., the
voltage rating of the energy source) and the current flowing
through the circuit. The relationship between power, current
and electric potential difference can be derived by
combining the mathematical definitions of power, electric
potential difference and current. Power is the rate at which
energy is added to or removed from a circuit by a battery or
a load. Current is the rate at which charge moves past a
point on a circuit. And the electric potential difference
across the two ends of a circuit is the potential energy
difference per charge between those two points. In equation
form, these definitions can be stated as

Equation 3 above can be rearranged to show that the
energy change across the two ends of a circuit is the
product of the electric potential difference and the charge
- V
• Q. Substituting this expression for energy change
into Equation 1 will yield the following equation:

In the equation above, there is a
Q in the numerator and a
t in the denominator.
This is simply the current; and as such, the equation can be
rewritten as

The electrical power is simply the product of the
electric potential difference and the current. To determine
the power of a battery or other energy source (i.e., the
rate at which it delivers energy to the circuit), one simply
takes the electric potential difference which it establishes
across the external circuit and multiplies it by the current
in the circuit. To determine the power of an electrical
device or a load, one simply takes the electric potential
difference across the device (sometimes referred to as the
voltage drop) and multiplies it by the current in the
device.

Check
Your Understanding

1. The purpose of every circuit is to supply the energy
to operate various electrical devices. These devices are
constructed to convert the energy of flowing charge into
other forms of energy (e.g., light, thermal, sound,
mechanical, etc.). Use complete sentences to describe the
energy conversions which occurs in the following
devices.

a. Windshield wipers on a car

b. Defrosting circuit on a car

c. Hair dryer

2. Determine the ...

a. ... current in a 60-watt bulb plugged into a
120-volt outlet.

b. ... current in a 120-watt bulb plugged into a
120-volt outlet.

c. ... power of a saw that draws 12 amps of current
when plugged into a 120-volt outlet.

d. ... power of a toaster that draws 6 amps of current
when plugged into a 120-volt outlet.

e. ... current in a 1000-watt microwave when plugged
into a 120-volt outlet.

3. Your 60-watt light bulb is plugged into a 110-volt
household outlet and left on for 3 hours. The utility
company charges you $0.11 per kiloWatt•hr. Explain how
you can calculate the cost of such a mistake.

4. Alfredo deDarke often leaves household appliances on
for no good reason (at least according to his parents). The
deDarke family pays 10¢/kilowatt-hour (i.e.,
$.10/kW•hr) for their electrical energy. Express your
understanding of the relationship between power, electrical
energy, time, and costs by filling in the table below.