### Video Transcript

In this video, weโre learning about
the power of electrical components. This kind of power is not always
the easiest thing to notice. After all, electrical components โ
like a computer or a lamp or a microwave โ are fairly stationary devices. Mechanical power involving fairly
large objects in motion may be more familiar to us. But that doesnโt mean that
electrical power isnโt a very real phenomenon.

We can start our discussion of
power for electrical components by recalling the definitions of a number of
terms. For starters, we can remember that
energy is defined as the ability to do work. This work that weโre talking about
could be mechanical work such as moving a mass up a hill or it can be electrical
work. An example of electrical work would
be moving a charge โ say this one over here โ while itโs in the presence of an
electric field. The analogy would be moving a mass
in the presence of a gravitational field. To do this, it takes work. And that work is the measure of the
energy invested in the process.

Moving on, letโs now recall the
definition for power. Power is defined as the amount of
energy transferred over some amount of time written as an equation. We can say that power ๐ is equal
to energy ๐ธ divided by time ๐ก. And at this point, letโs recall
that energy โ as we saw earlier โ is the ability to do work. Going back to our positive charge
in an electric field, letโs say we did some amount of work on this positive
charge.

Letโs imagine we did work ๐ in
moving it towards the other positive charge given off the field. In doing that much work measured in
joules, weโve exerted that much energy also measured in joules. In other words, for this process,
we can rewrite the energy used in terms of the work done. We could say that in this case the
power is equal to the work weโve done in moving the electric charge divided by the
time it took to move it.

As weโve said, weโre working with
an electric charge. And letโs say that this charge has
a charge ๐. What we can do now is come back to
our equation for power and multiply both the numerator and denominator on the
right-hand side by that charge ๐. It will become clear in a minute
why weโre doing this. But for now, just notice that by
multiplying by ๐ divided by ๐, weโre effectively multiplying by one; that is,
weโre not changing the equation.

So we have power is equal to work
divided by time multiplied by charge divided by charge. And as a last little manipulation,
letโs switch the denominators here. Letโs switch the ๐ and the ๐ก,
which algebraically we can do. Now that we have this equation for
power in this form, letโs leave it alone for a moment and go on with our
definitions. Next up is voltage.

Voltage โ also called electrical
potential โ is equal to electrical potential energy per unit charge. What on earth does that mean? Well, letโs consider it in the
context of our electric charge in our electric field. As we mentioned, this is a bit
analogous to a mass in a gravitational field. And if we think of it that way, it
can be helpful. Right now, this electric charge has
a tendency to move. Thatโs because itโs in an electric
field.

We can see that that tendency to
move โ how much this charge wants to move so to speak โ is a measure of its electric
potential energy. And notice that thatโs a lot like
the gravitational potential energy of a mass in a gravitational field. So anyway, this charge ๐ has an
electric potential energy. And we can refer to it for short as
EPE. If we were to take this electric
potential energy that the charge has by virtue of being in an electric field and
divide it by the amount of charge ๐ that the charge possesses, then what our
definition for voltage is saying is that this fraction is equal to electric
potential or another word for that is voltage.

But now take a look at this, in the
numerator of the left side, we have an energy. And as we saw earlier, energy is
the ability to do work. In fact, the electric potential
energy of this charge here is equal to the work it would take to bring the charge to
this particular location from infinitely far away. We can say that this is the same
amount of work capital ๐ as we referred to in our equation for power. That just as well could be the work
done to bring the charge in from infinitely far away to its current position.

If we make this substitution
replacing EPE with the work done on the charge, we see something interesting in
terms of the definition for voltage. We see that voltage or equivalently
electrical potential is equal to the work done on a charge divided by the amount of
charge it possesses. And notice this, just as in this
equation, we see work divided by charge. So in our equation for power, we
have a ๐ divided by ๐ term. That means in the equation for
power, we can replace ๐ divided by ๐ with ๐ voltage.

Now having done that, letโs move on
to our last definition: the definition of current. Current is defined as the amount of
electric charge passing a point over some amount of time. Writing this as an equation, we can
say that ๐ผ current is equal to charge ๐ divided by time ๐ก. And this definition is quite useful
because notice up in our equation for power that we have a ๐ divided by ๐ก
term. In other words, we can replace that
term with ๐ผ, the current.

With this substitution made, we now
have our equation for power for an electrical component. Itโs often expressed this way: ๐
is equal to ๐ผ current times ๐ voltage. There are a couple of helpful
things to notice about this equation. And to see them, letโs a bit of
space on the bottom of our screen.

Okay, the first thing to notice is
that according to our definition for power, power is equal to some amount of energy
transferred over some amount of time. That means that energy divided by
time is equal to current times voltage. And if we then multiplied both
sides of the remaining equation by time, we see that term cancels on the right-hand
side. And we have an equation that says
that energy is equal to time times current times voltage.

And now we can remember from
earlier our definition of current that current is equal to charge divided by
time. This means we can substitute ๐
divided by ๐ก in for ๐ผ in this equation. And notice what happens when we
do. The factor of time ๐ก cancels
out. And we then have an equation for
energy, which says that itโs equal to charge multiplied by voltage. So once we arrived at the
expression power is equal to current times voltage, we were able to use that to find
this expression for electrical energy that itโs equal to charge times voltage.

But there is a second thing we can
do with this power equation. Ohmโs law says that if we have a
resistor of constant resistance value then we multiply that resistance by the
current running through it, then that product is equal to the potential difference
across the resistor. Letโs say we were to take Ohmโs law
and multiply both sides of the equation by the current ๐ผ.

If we did that, then the left-hand
side of this expression would be equal to ๐ผ times ๐ which is equal to electrical
power, which means that the right-hand side of this expression is an equivalent way
to write electrical power: ๐ผ times ๐
times ๐ผ or ๐ผ squared ๐
. So then, not only is electrical
power equal to ๐ผ times ๐, itโs also equal to ๐ผ squared times ๐
. And perhaps, you can see thereโs
even another way to write electrical power. We could also write it as ๐ which
is equal to ๐ผ times ๐
by Ohmโs law all squared divided by the resistance ๐
.

So what we found then is several
ways to express electrical power and one way to express electrical energy. And weโve seen that these equations
come from basic definitions of electrical quantities backed up by simple charge
motion scenarios. Now that we know these equations,
letโs get a bit of practice using them through a couple of examples.

An electric motor is connected to a
nine-volt battery. Over a period of time, the motor
converts 450 joules of electrical energy into kinetic energy, heat, and sound. How much charge passes through the
motor over this period of time?

So what we have here is an
electrical motor being powered by a nine-volt battery. We want to know over the time it
takes the motor to convert 450 joules of electric energy into these other kinds of
energy โ kinetic energy, heat, and sound โ we want to know how much charge passes
through the motor over that time. To figure this out, we can recall
the relationship that connects voltage, energy, and charge.

Electrical energy ๐ธ is equal to
the amount of charge ๐ multiplied by the potential difference across which the
charge moves ๐. In our particular case though, we
donโt want to solve for ๐ธ, but we do want to solve for the charge ๐. So we can rearrange this
equation. When we do, we see that charge ๐
is equal to energy divided by voltage. And in our problem statement, weโre
told the energy used by the motor as well as the voltage power in it.

When we substitute in these values,
we have 450 joules of energy divided by nine volts of potential difference. This fraction comes out to 50
coulombs of charge. This is the amount of charge that
passes through the motor over this period of time.

Now, letโs look at a second
example.

The diagram shows a circuit
consisting of a light bulb connected to a cell. The potential difference across the
bulb is nine volts, and the current through it is four amps. How much is the power of the light
bulb?

Taking a look at this circuit, we
see that indeed this bulb is set up in series with a cell. With the circuit set up like this,
the bulb will be on shining light and we want to know just how much power itโs using
up as it does so. Weโre told the potential difference
across the bulb as well as the current through it. And we can recall the relationship
for potential difference current and power. That equation tells us that power
๐ is equal to current times voltage.

Applying this to our scenario, we
can substitute in the given values of current four amps and voltage nine volts. Then when we go and multiply these
quantities together, we find the result of 36 watts. Thatโs the power of the light bulb,
likely given off both in the form of light as well as heat energy.

Letโs summarize what weโve learned
so far about the power of electrical components.

In this section, we learned that
electrical power is given by the relationship ๐ is equal to ๐ผ times ๐, current
times voltage. We also saw that thanks to Ohmโs
law there are equivalent ways to write this expression. We can write it as ๐ผ squared times
๐
or we can write it as ๐ squared divided by ๐
. These are all equivalent
expressions for electrical power.

We also saw that electrical energy
๐ธ is equal to charge multiplied by voltage. Specifically, this is the charge ๐
that passes a certain point โ say in an electrical circuit โ multiplied by the
potential difference supplying that circuit. Furthermore, we saw that mechanical
processes such as lifting up a mass or climbing a set of stairs can help us
understand and clarify electrical terms and phenomena, such as voltage, electrical
potential energy, and power.