# Lesson Video: Efficiency Physics

In this video, we will learn how to calculate the efficiency of electrical devices given their useful energy output and total energy output.

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### Video Transcript

In this video, our topic is efficiency. In particular, we’ll be talking about electrical efficiency. This describes how well electrical components and devices use the energy that’s supplied to them. To see what a difference efficiency can make, consider these two electrical circuits we see on screen.

Let’s say that as far as the wiring and the power supply goes, these two circuits are identical. And what’s more, let’s say that the power rating of the bulbs that we put in these circuits is the same. So maybe they’re both 50-watt bulbs or 75-watt or some value like that. The point is, the power ratings are the same, but the bulb types are different. In the circuit on the left, we have an incandescent light bulb and, in the circuit on the right, we have an LED bulb. Considering that these two different bulbs are plugged in to identical circuits and they have the same exact power rating, we might expect the same amount of light to come from each one. But what we find is that much more light comes from the LED because it’s much more efficient than the incandescent bulb.

Efficiency has to do with how much output we get from something given how much input it receives. Let’s say that this box is some electrical component or some electrical device. It has the capacity to have energy transferred to it and then to use that energy to perform some function or do some task. Up top, we have the energy being input into this device, and then this device or this process creates some output. Thinking back to our light bulbs, the input was electrical energy and the output was radiant energy.

For any type of device, that device’s efficiency has to do with comparing its energy output with its energy input. When we describe energy input, that’s the total energy that goes into some device in order to create some result. We could describe this as the total energy supplied for this process. And then when we think about outputs, we’re specifically talking about useful energy outputs. Now, what we describe as useful depends on our particular application.

But think back to our incandescent light bulb. When this bulb is lit, not only does it give off light, radiant energy which we want and consider useful, but it also gives off lots of heat energy, which is not desired when we want to light a room. Most of the time then, the useful energy given off by a light bulb is its radiant energy, the energy it emits as light. And typically, the heat energy that this bulb gives off is considered wasted or not useful energy. Efficiency has to do with the total energy that goes into a device or a process and the useful energy that’s output from it. It’s expressed as a ratio, the useful energy output by a device or a process to the total energy that was input.

We say that this ratio, this fraction, is equal to the efficiency. And we use the Greek letter 𝜂 to represent that term. Now, looking at this relationship, let’s consider the highest and lowest efficiencies we could possibly have. Let’s say that there was some process where all of the energy input was output as useful energy. In other words, the useful energy output equaled the total energy input. In that case, this fraction would be equal to one. So one is the highest possible efficiency in decimal form that something can have.

And then at the other end of the spectrum, say that we had a process where some amount of energy was input but that we had literally zero useful energy output from it. Thinking back to our light bulb example, perhaps this would be a light bulb that didn’t give off any visible light when it was powered, but only heat. If the useful energy output was zero, then the numerator of this fraction is zero, and so the efficiency is zero too. So 𝜂, the efficiency of a process or device, is less than or equal to one and it’s greater than or equal to zero. For virtually all of the processes we’ll encounter though, 𝜂 is somewhere between these extremes. Here’s what that means.

For a given process where some total amount of energy is input, some of that energy is output in a useful fashion. And some of it is not. It’s energy that’s wasted. When our process or device delivers both useful energy output and wasted energy, then our efficiency will be somewhere between one and zero. Now sometimes, efficiency is expressed as a decimal value, like it would be in this case. But in other cases, it’s expressed as a percent. Describing some device, we might say that it’s 25 percent efficient or 80 percent efficient or something like that.

So it’s worth knowing how to go from a number written as a decimal — we’ll call that 𝑁 sub D — to an equivalent number written as a percent. All we have to do to convert a decimal number into its equivalent percent is to multiply that decimal by 100 percent.

For example, say that we calculated an efficiency of 0.03 written as a decimal. If we want to know what percentage efficiency that is, we multiply that by 100 percent and find a result of 3 percent. And then to see how to go the other direction from a number given as a percent to a number written as a decimal, if we divide both sides of this equation by 100 percent, then that factor cancels out on the left. And we see that, to make this conversion, we take our number, originally written as a percent, divide it by 100 percent. And that equals the equivalent value as a decimal.

So, for example, if we had an efficiency written as a percent of 37 percent, then to write that as a decimal, we would divide it by 100 percent. In doing this, we see that the percent symbol cancels out, and we get a result of 0.37. That’s 37 percent written as a decimal. One more thing to know about efficiency is that sometimes it’s not written as a ratio of energies, but as a ratio of powers.

Returning once more to our light bulb example, most light bulbs are rated in terms of the power that they consume. That’s the number we can see written on the box that the light bulbs are sold in. That number, whether it be 100 watts or whatever the power rating it is, represents the total power being input to that light bulb during normal operation. And then along these lines, we look at the useful power output by the bulb in terms of the radiant energy it gives off over some amount of time.

So sometimes when we’re calculating the efficiency of an electrical device or component, instead of referring to the useful and total energy, we’ll use numbers referring to the useful and total power instead. It will be clear from the context which form to use. And as a rule of thumb, the standard way of understanding efficiency is in terms of energies, the useful energy output to the total energy input. Knowing all this, let’s get some practice with efficiency through an example exercise.

An LED light has an efficiency of 29 percent. If it converts 5,800 joules to light, what is the total energy it was supplied with?

Okay, so we have an LED light. And that light has a given efficiency. This LED is supplied with some total amount of energy. And then it outputs a given amount of energy as light. And that given amount, we’re told, is 5800 joules. We can consider this to be useful output energy from the LED. And this useful energy output is some fraction of the total energy input into the device. Now, if our LED was perfectly efficient, then all of the energy we input to it would be converted to useful energy output as light.

But we know that that’s not the case. Rather, just under 30 percent of the input energy is converted into useful output energy. That’s what it means that the efficiency of this LED is 29 percent. So basically, we input some total amount of energy to our LED and then 29 percent of that total input is output as the 5800 joules of light energy. And knowing this, we want to work backward to solve for that total energy input.

Let’s let the total energy supplied to our LED be represented by this symbol, 𝐸 sub t. And then let’s let 𝐸 sub L represent the energy that our LED converted to light. The LED efficiency of 29 percent means that if we take 29 percent of 𝐸 sub t, then that will equal 𝐸 sub L. Written out using words, 29 percent of 𝐸 sub t equals 𝐸 sub L. And now, we can translate this into an equation that we can solve for 𝐸 sub t. To do that, we’ll want to convert this percent, 29 percent, into a decimal. In general, if we have a number written as a percent — we’ll call it 𝑁 sub P — then if we divide that value by 100 percent, that equals the equivalent number written as a decimal.

So if we take 29 percent and divide that by 100 percent, we see the percent signs cancel out. And as a decimal, this fraction is equal to 0.29. So if we take this number, 0.29, and multiply it by 𝐸 sub t, the total energy supplied to the LED, then that product will equal 𝐸 sub L, the energy output as light. Written as an equation, we can say that 0.29 times 𝐸 sub t equals 𝐸 sub L. And to solve for 𝐸 sub t, the total energy, we’ll divide both sides of the equation by 0.29. That will cancel that term out on the left. And we see that the total energy input is equal to the light energy, the useful energy output, divided by 0.29.

And we know 𝐸 sub L, the energy converted to light; that’s 5800 joules. So 𝐸 sub t is 5800 joules divided by 0.29 which is equal to 20000 joules. That’s the total energy supplied to this LED light.

Let’s now look at a second example exercise.

A 120-watt television has a useful power output of 30 watts. What is the efficiency of the television?

Okay, so we have this television, and the television is rated at 120 watts. That means if we were to monitor the power coming into this television during its normal operation, that power would be at 120 watts. So that’s the total power input. And we’re told that this television has a useful power output of 30 watts. That is, the power that it uses in support of creating a moving image is 30 watts. So that leaves 90 watts doing something other than what we want them to be doing. Most of that wasted power goes into generating heat in the system. So then our television usefully outputs some of the power input to it but not all of it. And we want to calculate its efficiency.

Efficiency can be represented using the Greek letter 𝜂. It’s equal to the useful output from a process or a device divided by the total input. These outputs and inputs can be energies or, like in our case, they can be powers. In this instance, we’re told that the useful power output from this television is 30 watts and that the total power input is 120 watts. We can see when we calculate this fraction that the units, watts, will cancel out. And we’re left with 30 divided by 120. Written as a decimal, this is equal to 0.25.

Often though, efficiencies are expressed as percents. To write 0.25 as its equivalent value as a percent, all we need to do is multiply it by 100 percent. 0.25 times 100 percent is equal to 25 percent. That’s the efficiency of this television. It means that 25 percent of the total power input to it is converted to useful power output.

Let’s now summarize what we’ve learned about efficiency. We saw in this lesson that the efficiency of a device relates its useful output to its total input. We also learned that, written as an equation, energy efficiency is equal to the useful energy output divided by the total energy input. And this efficiency is always less than or equal to one and greater than or equal to zero.

It’s worth pointing out though that no perfectly efficient devices or processes have yet been discovered. That is, practically speaking, 𝜂 is always less than one. We found out how to convert between a number written as a percent and a number written as a decimal. Multiplying a number written as a decimal by 100 percent converts it to a number expressed as a percent. Lastly, we learned that efficiency may describe energy, or it may describe power inputs and outputs. This is a summary of efficiency.