Welcome back. 
I'm Nathan Parrish. 
And again, this is linear circuits. 
Today, we're going to be talking about 
power and energy. 
So first of all, we're going to be trying 
to calculate power and energy. 
Describing the difference between power 
and energy. 
Using the conservation of energy to find 
an unknown energy. 
And using power to calculate an unknown 
current or unknown voltage. 
So from our previous class, we talked 
about voltage, this kind of this electric 
potential based upon the distribution of 
charge inside of a device. 
And then we looked at the battery. 
And how it could be charging or 
discharging, based upon the direction of 
the current flow. 
But in the same context, now we can start 
talking about power and energy. 
And, in the next class, we will look at 
some basic circuit diagrams. 
But actually today, we're going to be 
doing our first bit of circuit analysis. 
Lesson objectives are to calculate power 
from an energy function,and to calculate 
energy from the power function. 
To use the conservation of energy to find 
a power of an unknown device, calculate 
power from voltage and current, and 
finally to find a voltage or a current 
for a device with an unknown, for a known 
power. 
So in talking with, about power, power is 
measured in units of watts and power is 
essentially the rate at which energy is 
being consumed. 
Energy being measured in joules. 
So power is in joules per second. 
And one watt is equal to one joule per 
second. 
Now, we can actually use a little bit of 
calculus to help us do some calculation 
about power. 
Power is going to be the change in energy 
with respect to time. 
Or we can use the chain rule, this sort 
of, the change in energy with respect to 
charge times the change in charge with 
respect to time, is going to be 
equivalent. 
And so, if we have energy divided by 
charge, you might remember from last 
class, that's voltage. 
And if I have charge with respect to 
time, that's going to be current. 
So it turns out that we can calculate 
electric power by taking a voltage and 
multiplying it by a current. 
We can also find an energy by integrating 
the power with respect to time, since 
power is the rate at, at which energy is 
being consumed or generated. 
So all we have to do is integrate with 
respect to time, and then we can find our 
energy. 
In calculations, we're going to be using 
w to represent energy. 
And we're going to use p to represent 
power. 
One of the things that students sometimes 
forget that's actually a very useful tool 
when you're doing analysis, is 
remembering that energy cannot 
instantaneously change. 
The power can, it's not a problem. 
So we can look at this example. 
Here we have an energy curve, where the 
energy is being consumed, the energy 
decreases and decreases and decreases 
until this point, where there's no energy 
left. 
And now, it is regenerated. 
And remembering that we can calculate 
power by taking the derivative of energy. 
But we know that this derivative is 
basically the slope of the slide and so 
we are basically just going down one. 
So oue power is negative 1 watt. 
Over here the slope is now positive 1. 
So we have 1 watt. 
At this point in the middle, it's kind of 
a mathematical problem. 
There is no derivative here because if we 
come from one side or the other, we get 
two different answers. 
And so it's kind of undefined what the 
derivative happens to to be here. 
But this only happens at one isolated 
point, and so it really doesn't matter, 
as far as we're concerned. 
So here we just used some circles to 
identify that we don't have a defined 
amount of power. 
Now if we take an integral, of the power, 
we get the energy. 
Now suppose that we wanted this energy, 
to change, instantaneously in time. 
Or just do jumps. 
What would have to happen? 
Well, over a tiny little slice, if we 
remember the definition of an integral, 
we have to have an inordinately large 
amount of power. 
So large, in fact, that as we make the 
area around it smaller and smaller, the 
value has to overcome that ever 
decreasing window of space. 
Which basically means that we have to 
have infinite power. 
Well infinite power is not something that 
we can have. 
So it turns out that we cannot have the 
energy change instantaneously. 
But power can change instantaneously as 
much as it likes. 
Now suppose that you run a power company 
and you want to charge your customers. 
Well, how are you going to charge them? 
Are you going to charge them by the 
amount of power that they are using or 
the amount of energy? 
And why? 
I want you to think about it for a 
minute. 
In fact, I'm going to even stick up a 
little sign. 
A little pause, to let you know that it's 
okay to pause the video and take a little 
bit of time to think about it before you 
answer. 
So what did you come up with? 
It turns out, if you look at your power 
bill, you're probably going to see 
something about your power company 
charging you in units of kilowatt hours. 
And this is actually a measurement of 
energy. 
It seems to be a little bit easier to 
work with than joules, since a kilowatt 
hour is equal to about 360 joules. 
And so it's just an easier number to work 
with. 
But essentially, it's energy. 
And why energy? 
Well that's because they don't really 
care about what rate you're using the 
power. 
What they want to know is how what, or 
how much energy you're actually 
consuming. 
So it's important for them to keep track 
of that. 
But we shouldn't lose complete identity 
of power because it could become a 
problem there as well. 
Suppose that everybody in the city 
simultaneously decided to run their 
vacuum cleaners and air conditioners all 
at once. 
Well, the power company might have some 
problems. 
They wouldn't be able to generate enough 
power that instant to be able to provide 
for everybody at once. 
Even though the energy wasn't too much 
for it to provide, it's the rate that 
becomes a problem. 
So, in practice, your power company kind 
of looks at both. 
But when you're charged, you're going to 
be paying for energy. 
Another very useful property that we can 
use, is something called the conservation 
of energy. 
Just like most closed systems, the total 
amount of energy that is being provided 
or consumed by the system, is a constant 
value, in time. 
No energy is created or destroyed in the 
system. 
But, we actually are going to find it 
more convenient, generally, to work with 
power rather than energy. 
And so let's take a look at a little bit 
of a derivation here. 
We sum up the total energy in a system. 
We're going to get some constant value. 
Now what we can do then is take the time 
derivative of this value and if I take a 
time derivative of some constant, that's 
going to be equal to zero. 
And it turns out that as long as this sum 
is over a finite number of energies, we 
can interchange the derivative and the 
sum to give us this, where we're summing 
up the derivatives. 
Well these derivatives are just powers, 
individual powers, and we're summing all 
of these powers up and we then get zero. 
So what does this actually mean in 
practice? 
Well, what it really means is that 
because energy is conserved, we know that 
the rate of power which a system is 
consuming or generating energy, has to be 
0 overall, throughout the system. 
So if we sum up all of the powers that 
are being consumed and all of the power 
that's being generated in the system at 
any point in time, it always has to be 0. 
So this is the excellent way to check 
your work. 
It also turns out to be something useful 
on analysis, if we happen to know 
something about the power in the system. 
Again reference directions are going to 
come up. 
like I said, it's something that students 
really have challenges with, especially 
at first, keeping the reference 
directions straight. 
And because we're taking a product of a 
current and a voltage, if we just naively 
multiply them together, we can get 
different answers depending upon how 
they're related. 
And as far as we're concerned, there's 
only two real configurations to consider. 
The first configuration is where our 
current arrow is pointing from our plus 
to our minus. 
And the other is when our current arrow 
is going from our minus to our plus. 
So, then we'll start off by looking at 
the one on the left, where the arrow goes 
from the plus to the minus. 
We're going to calculate power by 
multiplying the current and the voltage. 
What this ends up doing then, is it means 
that positive power means power's being 
consumed, not generated. 
This might be counterintuitive to what 
you might think. 
Thinking that negative would be that it's 
being used and positive would be that 
it's being generated. 
But we're actually going to have positive 
power being consumed power. 
And that's the general trend that we're 
going to use. 
And so consequently, if we flip the 
references, where the arrow's going the 
other way, to keep the values consistent, 
we have to multiply by a negative 1. 
So in this case, power is going to be 
equal to minus i times v, minus the 
current times the voltage. 
So be very careful when you're 
calculating power to look. 
Is my arrow going from the plus to the 
minus? 
Just multiply them together. 
Is it going from the minus to the plus? 
Remember to switch to the reference 
directions so you're multiplying by 
negative one. 
So now we're going to look at a 
practical, practical problem, and do an 
analysis. 
Now suppose we have a light bulb. 
And we have 120 va, volts, of voltage, 
going to the light bulb to power it. 
And this particular light bulb is rated 
60 watts. 
Now suppose we want to know the current 
that is flowing through the light bulb. 
Well at this point we have all the tools 
we need to be able to solve this problem. 
So, I want you to take a look at the 
system and try and solve it yourself. 
And so I'm going to go ahead and put up a 
pause button and then we will continue on 
after you've had a chance to try to solve 
it yourself. 
Okay, so we have 60 watts of power. 
And we notice that the arrow for current 
is going from the plus to the minus. 
So in this configuration, power p is 
equal to i times v, current times 
voltage. 
So that means the power, 60 watts, is 
equal to 120 volts times i. 
Where i is equal to 60 watts divided by 
120 volts. 
So the current is equal to one half of an 
m. 
Now, if you were able to solve this 
problem, it might be a good thing to go 
onto the forums and post, because you'll 
notice that we're not going to actually 
show the answer to the solution here on 
the slide. 
To summarize, we've describe the 
relationship between power and energy and 
how to calculate them. 
We also looked at how voltage and current 
interact to give us electric power, and 
in turn, electric energy. 
We presented a derivation for the 
conservation of power, and how this 
property can used in analysis. 
And we solved our very first simple 
analysis problem. 
Next lesson, we will be taking our first 
look at circuit diagrams and seeing how 
these things come together. 
I remind you again, to take a look at the 
forms. 
Answer questions that you know the 
answers to. 
It's great practice, teaching others. 
As well as ask any questions that you 
might have from this lesson. 
Until then, we will see you next time. 
Cheers.