Video: Power and Energy

In this lesson, we will learn how to use the formula 𝑃 = 𝐸/𝑡 to calculate the rate at which energy is transferred by a device given the amount of energy it transfers in a given time.

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Video Transcript

In this lesson, we’re learning about power and energy and our focus is from analytical perspective. We’ve seen how these terms relate to mechanical processes such as pushing a stone up a hill or climbing a set of stairs. Here, we’re going to see how to work with power and energy in an electric circuit or an electric appliance. Let’s start off by recalling a couple of definitions.

First, the definition of energy, energy is the ability to do work. For example, if you were to start at the bottom of a staircase and then climb to the top of it, that would mean you had supplied the energy to do some amount of work against the gravitational force pulling you down. Notice that this definition for energy has nothing to do with how long it would take for you to get from the bottom of the staircase to the top. In fact, that’s where power comes in.

Power is defined as the amount of energy used over a certain time. So for example, if you were to climb up these steps two times, the first time taking five seconds to do it and the second time taking half that amount 2.5 seconds, then even though in both cases you would use the same amount of energy, you wouldn’t use the same amount of power. In the first case when climbing the stairs took five seconds, you would have used half as much power as you used when it took 2.5 seconds. The waiver in this definition for power in words often is expressed as a mathematical equation that the power 𝑃 is equal to the amount of energy used 𝐸 divided by the time elapsed 𝑡.

Considering the units in this expression, we know that the units of energy are joules symbolized by a capital 𝐽. By the way, that’s the same unit as the unit for work, showing how closely energy and work are related. The base unit for time is seconds and the unit for power we know is watts symbolized by capital 𝑊. We can see from this equation that one watt is equal to one joule per second.

So far, all of this discussion has had to do with a mechanical form of energy and a mechanical form of power. But of course, energy and power can exist in electrical form as well. Just think of any electrical appliance you might have, like a toaster or a television or a microwave. All of these and every other electrical appliance uses some amount of energy and it uses that energy over some time interval. Therefore, it has a power consumption rate.

When we think about energy and power in an electrical context, the question may come up just what is electrical work or what does work look like for an electrical system. To understand that, we can think back to the basics of electricity: positive and negative charges. We know that oppositely charged particles will attract one another and that particles with a similar charge will repel one another. So that’s a natural tendency. Now, let’s say we try to resist that natural tendency or work against it.

For example, let’s say we took this positive charge here that wants to move off to the right under the repulsive influence of this other positive charge. We’ll see we take this first charge and instead we start to push it towards the left. If we pushed hard enough, indeed we could close the gap between these two positive charges. If we did that, we’ll be going against the natural inclination of these charges to push one another apart.

Going back to our picture on the opening screen of a person pushing a stone up a hill, this is a bit like what it would be like to push this positive charge towards the other positive charge. The charge just like the stone wants to move in a certain direction due to the force field that it’s in. But we resist to that motion and in fact push it the opposite way. We do work.

If we were the ones doing the work on this positive charge, then we could say that as it gets closer to the other one, we do positive work. And as it moves away from the other one, we do negative work and this work is a measure of the energy that we’re putting into the system. And if that energy for moving charge is released over some amount of time, then we have electrical power.

One place that we may have seen electrical power reference before is if we’ve ever gone to buy a box of light bulbs, usually the power rating of the bulb is printed on the box. This power rating shows how much electrical energy the bulb consumes over time and corresponds to the brightness of the bulb. Knowing all this, let’s get some practice now working with a few examples of electrical power and energy.

A 60-watt incandescent light bulb is left on for 30 seconds. How much energy is supplied to it over this time?

We have this light bulb then that uses up 60 watts of power. If we recall that power is equal to energy divided by time and that the units of power are watts, the units of energy are joules, the units of time are seconds, then we can see that the fact that this is a 60-watt bulb means that it uses up 60 joules of energy every second.

We can see that this way. Let’s say we rearrange this power equation so that energy is isolated on one side by itself. When we do that, we see that energy is equal to power times time. Substituting in for those values, we see it’s equal to 60 watts multiplied by the time the bulb is on, 30 seconds.

But look at this: power is equal to energy divided by time, which means that one watt is equal to a joule per second. This means we can go to our expression for power 60 watts and rewrite it as 60 joules per second. That’s because a joule per second is a watt. Once we’ve done that though, we see that the units of seconds in the denominator cancel with those in the numerator. We’re left in this expression with the units of joules, the units of energy.

So the energy 𝐸 then is equal to 60 joules multiplied by 30. 60 times 30 is 1800. So 1800 joules is the amount of energy supplied to the bulb over this time. And we found that result by multiplying the power used by the bulb times the time it was on.

Now, let’s look at a second example of electrical energy and power.

A microwave is used for five minutes, during which time it is supplied with 180 kilojoules of energy. What is the power of the microwave?

Okay, so we have this microwave. And clearly, something delicious is cooking inside. It’s being used for five minutes. We’re told that over this span of time, 180000 joules of energy is delivered to the microwave. The question is what’s the power of the microwave. To see what this is, let’s recall the relationship between power, energy, and time.

The equation connecting those three says this. It says that the power involved in a process is equal to the energy used divided by the time that the process took. In this scenario, we’re given both the energy used and the time elapsed. So we can substitute in those values now. The power of the microwave is equal to 180000 joules divided by five minutes.

Now, what we’ve written so far is true. But the units in this expression, particularly, on the right-hand side, aren’t what we want them to be. We can recall that the base units of energy are joules, the base units of time are seconds, and the base units of power are watts. In other words, we can see that one watt is equal to one joule of energy used up in one second.

When we go to calculate the power of the microwave, we would like to get our answer in units of watts. But in order to get that, we need to have the right-hand side in units of joules per second. We haven’t quite got there yet. We can see that our energy is currently in units of kilojoules not joules and our time is in minutes rather than seconds. Here’s what we can do about that.

We can recall in the first place that one kilojoule is equal to 1000 joules. That’s what the prefix kilo refers to. And we can also recall when it comes to time that one minute of time is equal to 60 seconds, where seconds of course is the time unit we’re interested in. If we take this information then and apply it to our fraction, we can write that our power is equal to the energy of 180 times 10 to the third joules divided by the time of five times 60 seconds.

And notice now that we’ve done this conversion, we have units of joules per second, in other words watts. When we calculate all this out, we find an answer of 600 watts. So that is the power of this microwave.

Let’s take a moment now to summarize what we’ve learned about power and energy.

First off, as a recap, we saw that energy is the ability to do work and it’s measured in units of joules, abbreviated with a capital 𝐽. We further saw that power is energy used per unit time and it’s measured in units of watts. The relationship we saw between power and energy is that power is equal to energy divided by time. And we also saw through our examples that it’s important to keep track of the units: joules for energy, seconds for time, and watts for power.

And we also saw that electrical power and energy has to do with positive and negative charges and the forces between them, either attractive or repulsive. So just like in mechanical systems, power and energy apply to electrical systems as well. The situations are analogous. And the equation that connects power and energy is the same in each case.

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