Good morning.
Today, we are going to derive equations for two typical arrangements of resistors; resistors
in series and resistors in parallel.
[Intro]
We start with a circuit with a battery and two resisters in series.
Because a positive charge would be repelled from the positive terminal of the battery
and attracted to the negative terminal of the battery, the current in this circuit is
clockwise or up through the battery and down through each resistor.
Let’s label those currents as the terminal current through the battery and current 1
and current 2 through their respective resistors.
Hopefully you recognize that each charge on the wire has to go through all three of these
circuit elements, therefore, all of these currents are equal.
According to Kirchhoff’s Loop Rule, a charge moving all the way around a loop in a circuit
must end with the same electric potential energy it started with, therefore, the electric
potential difference all the way around a loop is equal to zero.
If we define the loop in a clockwise direction in our circuit, Bobby, please tell me what
Kirchhoff's Loop Rule equation will be?
- Well, we know the electric potential difference around the loop equals 0.
As the loop goes across the terminals of the battery, it goes from the negative terminal
to the positive terminal therefore the electric potential difference across the battery equals
the positive terminal voltage.
As the loop goes across each of the resistors, it goes in the same direction as the current
across the resistors, therefore we subtract the electric potential difference for both
resistors.
And we get the terminal voltage of the battery equals the sum of the electric potential differences
across the resistors.
- Thanks, Bobby.
Now, because electric potential difference equals current times resistance, we can substitute
current times resistance for each of the electric potential differences.
For the battery, the terminal voltage equals the current at the terminals of the battery
times the equivalent resistance of the electrical load.
The electrical load in this circuit is the two resistors in series.
Therefore, the equivalent resistance in this circuit will be the resistance of the two
resistors in series.
And...
- Everybody brought "I" to the party.
- I think you mean everybody brought "current" to the party.
- Yeah.
- But those are three different currents.
- Remember, all the currents are the same.
- Oh, yeah.
That's right, I forgot.
Shall we?
- Okay.
- Everybody brought current to the party.
Everybody brought mass, mass.
- Therefore, the equivalent resistance for two resistors in series is the sum of those
two resistances.
We could perform this experiment with as many resistors in series as we wanted to and the
equivalent resistance would always be the sum of the resistances.
So, the equivalent resistance of resistors in series equals resistance 1 plus resistance
2 plus resistance 3 plus as many resistors as we have in series.
Please remember that in series circuit elements currents are the same and electric potential
differences add.
- When circuit elements are in series, their currents are the same, however, their electric
potential differences add.
Got it.
- Yap.
- I do want to clarify that the term "equivalent resistance" means all the resistors in series
in this particular case, could be replaced with a single equivalent resistor without
affecting the current or potential differences in the rest of the circuit because it has
an equivalent resistance.
- Okay.
- Now, let's do a circuit with a battery and two resistors in parallel.
Again, the current directions are up through the battery and down through each of the resistors.
There are two junctions in the circuit; junction a and Junction b.
Now, we know charge is conserved at a junction.
In other words, every charge that goes into the junction must come out of the junction,
which is what Kirchhoff's Junction Rule states.
Using Kirchhoff's Junction Rule on Junction a gives us the terminal current through the
battery equals the current through resistor 1 plus the current through resistor 2.
We can then return to Kirchhoff's Loop Rule, however, we need to first define the directions
of the three loops in this circuit.
Let's define all three as clockwise.
Loop A contains the battery and resistor one, loop B contains resistor one and resistor
two, and loop C contains the battery and resistor 2.
Billy, please apply Kirchhoff's Loop Rule to loops A and C.
- Okay.
The electric potential difference around loop A equals zero which equals the positive electric
potential difference across the battery, positive because the electric potential goes up in
the direction of the loop.
And then, minus the electric potential difference across resistor 1 that is negative because
the electric potential difference across a resistor goes down as you go in the direction
of the current across the resistor.
Therefore, the terminal voltage equals the electric potential difference across resistor 1.
For loop C... actually, the only difference between loops A and C is that C has resistor
2 in it instead of resistor 1, so the equations are the same just change the subscripts from
1 to 2.
- Thank you, Billy.
Notice then that all of the electric potential differences in this circuit are equal.
And because electric potential difference equals current times resistance, current equals
electric potential difference divided by resistance.
Bo, please combine these equations to solve for the equivalent resistance of the two resistors
in parallel.
- Sure.
Let's go back to our junction rule equation for junction a.
Terminal current equals current one plus current two.
We can substitute electric potential difference divided by resistance for each current.
For the resistors, we just add number of subscripts.
But for the battery, the subscripts are t for terminal voltage and eq for the equivalent
resistance of the electric load on the circuit or the two resistors in parallel.
So...
- Everybody brought electric potential difference to the party.
- Because all the electric potential differences are the same.
- Right.
Everybody brought mass, mass.
- Correct, Bo.
Now, we can take the inverse of the whole equation.
And because we could perform this with as many resistors in parallel as we want, the
equation for the equivalent resistance of resistors in parallel is the equivalent resistance
equals the inverse of the sum of the inverses of the resistances.
- The inverse of the sum of the inverses of the resistances?
- Yes.
The equivalent resistance for resistors in parallel equals the inverse of the sum of
the inverses of the resistances.
- Sure.
- Please, remember that in parallel circuit elements, currents add and electric potential
differences are the same.
- Right.
This is the reverse of what it is for series elements.
When circuit elements are in series, their currents are the same, however, their electric
potential differences add.
But for circuit elements which are in parallel, their currents add and their electric potential
differences are the same.
That make sense?
- Yap.
- Also, notice that adding resistors in series increases the equivalent resistance of the
resistors, and adding resistors in parallel decreases the equivalent resistance of the
resistors.
Think of it this way, by adding a resistor in series, you are adding resistance to the
path the charges have to go through which no matter how small that resistance is which
you are adding in series, that still increases the equivalent resistance.
When you add a resistor in parallel, you are adding an additional path for the charge to
go through and therefore, no matter how large that resistance is, which you are adding in
parallel, the addition of another pathway for the charges to travel through decreases
the equivalent resistance.
- Wait, so the equivalent resistance of a series circuit is always greater than the
largest resistance in the circuit.
- And the equivalent resistance in a parallel circuit is always less than the smallest resistance
in the circuit.
- That is cool.
- Now, it may seem pretty obvious in simple circuits like the ones we just went through.
However, it is important to identify when circuit elements are in series or in parallel,
and also to realize that sometime they are actually in neither.
Okay.
Let's start with series; if every charge that goes through one element also has to go through
the other elements those two circuit elements are in series.
For example, all the resistors in the following circuit are in series.
This is because every charge in the circuit has to pass through every one of the resistors
in this circuit.
- Seems pretty simple.
- But those five resistors which are drawn parallel to one another are all parallel to
one another.
- Yeah.
But that is just how the circuit was drawn.
Every charge has to go through all of the resistors so, all the resistors are in series.
- Yap.
- Okay.
- Correct.
And circuit elements which are all in parallel, all have the same electric potential difference.
For example, all the resistors in this circuit have the same potential at the top and bottom
of each resistor.
So, their electric potential differences are all the same.
Another way to look at this is that if the charges are split between resistors and then
all the charges come back together again, the resistors are in parallel.
- Still seems pretty simple.
- I am good.
- Good.
- If you look at this next circuit, it appears to be different.
However, the top of each resistor or right side in the case of resistor 4, are all at
the same electric potential and the bottom of each resistor or the left side in the case
of resistor 4, are at the same electric potential.
Therefore, all four of these resistors have the same electric potential difference and
are therefore still in parallel.
In fact, this is the same circuit as before, it has simply been drawn a little bit differently.
- Seems like we need to look carefully at each circuit to analyze which elements are
in series or parallel and not just depend on the geometry of the way the circuit looks.
- Yap.
- I agree.
Thank you very much for learning with me today.
I enjoyed learning with you.