So now that we have an idea about how we measure
different quantities in a circuit
as well as what Ohm's Law tells us about those quantities,
let´s talk a little bit more about the resistors that we've been discussing.
For given resistor so it might just be some pipe or sorry some wire,
maybe a copper wire like this one that we've zoomed in on very closely here.
We can ask the question, what is the resistance of the material itself.
So for example, what I mean here is that the resistance of this wire,
as electricity tries to go through it will depend not only on what material it is,
whether its copper or something else,
it also will depend on the geometry of the thing,
how big it is, how long it is and all of these things
will have an impact on how much resistance this wire gives us.
The question we're asking here is what if we don't want to talk about the geometry so much,
we're not so interested in how wide the wire is or how long the wire is,
we're just interested in the material like copper or whatever it might be.
So, for example, in this wire supposed we have some length, L
and some cross sectional area of our wire, A.
These are the kinds of parameters that the resistance will depend on,
but that we might not be interested in. And so we can write an equation,
for the resistance which turns out to look exactly like this,
that the resistance will be equal to the length of the resistor
divided by the cross-sectional area of the resistor,
basically telling us that the longer the wire is the more resistance there is,
and the bigger the cross-sectional area of the wire,
the less resistance there is since the current has more room to flow,
but they're proportional to the resistance by this Greek letter rho.
And this Greek letter rho is representing the resistivity of the material
which is just a measure having to do with the material itself.
In other words, this resistivity doesn't have anything to do with the geometry.
It doesn't care about how long the wire is, it doesn't care about how wide the wire is.
And so, for any material, like copper or silver or gold,
we can define some resistivity of that material,
that again will always be independent of the geometry of the particular resistor.
The resistor, as we've already said, when you have current flowing through it
will try to impede that flow, it´s going to lower the amount of pressure
that you have in your system by sapping the energy, that is going
the energy from the electrons or the current that is flowing through that resistor.
This energy has to go somewhere though,
it turns out that this energy is lost to heat,
so we can define the amount of energy per unit time,
that this resistor is dissipating or taking away from your system,
as a function of the current and the resistance.
And so this equation here, is a very, very common equation,
so, definitely be aware of what it is and what it means,
it is saying that the power which we've already talked about,
measured in unit of Watts, the energy per unit time being taken away
by a resistor is equal to the current that's going through that resistor square,
times the value of the resistance of the resistor,
and so we say power P equals I squared R the current squared time the resistance.
Again, be aware that the units of this power is exactly the same
as the units of power that we already introduced which is Watts,
or joules per second, how much energy are we using per unit time.
Also, be aware, that because of Ohm's Law we have a way to relate the voltage,
the resistance and the current, and so we could rewrite this power law,
P equals I squared R in a few different ways just by substituting variables using Ohm's Law.
Finally, remember that if we know how much energy per unit time is being used,
in other words, we know how much power is being eaten from your circuit by the resistor,
we can always find how much energy by multiplying by the time,
so for example, if a particular resistor was using 3 watts of energy for 2 seconds,
you could multiply these 3 watts by 2 seconds to see,
that the resistor was taking away 6 joules of energy in that duration in that period of time.
So now we're going to discuss how we can add resistors,
often in circuits we don't just have one simple resistor
we have many resistors and in a particular circuit that can be combined
in a few different ways, so we could add them up and think about in equivalent resistance.
So for example, supposed I have these two resistors R1 and R2
and the only thing the current can do is flow through both resistors.
It doesn't really have a choice here,
the way to find an equivalent resistance for both of this,
is to simply add them to something that we would call the total resistance,
the resistance of both resistors together.
In series, since these resistors would simply add as though they were one big resistor,
all we have to do is add the resistance of each individual resistor,
so nothing particularly complicated here,
this is actually the more intuitive of the equations for adding resistance
if you have them in series, you simply add the two resistances together.
And this is a very, very important point that is also very commonly asked on exams,
in an exam settings, so be very familiar with this equation.
As well as with the following one, which instead asks what if we have two resistors in parallel.
Parallel means that the current as it flows through the circuit,
does not have to go through both resistors,
but as it's coming from the left to the right, what it can do is pick a direction,
so it might go through to the first resistor or might go down and go through the second resistor instead.
So this is how we defined a parallel in a circuit,
if two things are in parallel what we mean, is that the current can pick which path to take.
When we're adding resistors in parallel,
and equated them to some total resistance, we have a different equation for adding them.
It's exactly the inverse of each term, so we would say one divided by the total resistance,
equals one divided by the first resistance, plus, one divided by the second resistance.
So we're going to use this equation a few times just so we can see how it works
and get into the practice of it, but certainly again, be aware
and understand the difference between the equation for adding resistors in series,
where the current has to go through both, or adding in parallel,
where goes to just one or the other.
So this summary slide gives us an idea of how both of this laws are working,
where again this series resistors act like one larger resistor,
whereas the resistors in parallel, are acting instead like a smaller resistor
or having less resistance because, if we solve this equation,
we will get a total resistance that was actually less than, either one of the two resistances.
One way to think about this, a way of getting less resistance by adding resistors in parallel,
is to think about the current that's flowing through your circuits sort of like traffic.
With more lanes, which would be like having resistors in parallel,
you actually get less resistance, because instead of the cars all having to go in the same direction
or through the same roads, you are giving them an option,
you are giving them more room and they are able to flow in different ways
and flow through either one of the resistors.
And so, this way of putting resistors together,
decreases the total resistance, rather than increasing it.
So again, we have Ohm's Law written here,
we have the voltage drop across a resistor,
is equal to the current through the resistor times the resistance in the resistor,
in each resistor in the circuit, so in this circuit I have two of them drawn here,
we'll drop the voltage by the current through that resistor,
times the resistance of that resistor. If we wrote down the voltage,
for every object in this particular circuit, we have to require,
which will discuss the physics of a little bit later,
that the entire voltage for the whole system added together, must be zero.
And when I say the whole system, what I mean is were also going to include the battery,
so the battery, in this case, is acting like a pump it's adding pressure,
or it's adding voltage to your system.
Whereas each resistor is taking more pressure from your system
by sapping the energy by forcing the current to go through
an object that is resisting its flow.
In so, if we added up all the resistance contributions or the voltage contributions.
What we have is a voltage added from the battery,
so we have plus V, but then we have voltage taking away
because the current is having to go through each resistor.
Using Ohm's Law, we know exactly how much voltage this is,
so we can write plus V for the battery, minus I times R for each resistor.
So, we have V minus I times R for the first resistor minus I times R for the second resistor
and we know again for physics reasons
we'll discuss shortly here at this total has to sum to zero.