00:01
We are to prove or derive the equation or formula for the equivalent resistance of resistors in series and equivalent resistance of resistors in parallel.
00:14
So we already have the diagrams here.
00:18
We have three resistors and we only need omslo to help us in our derivation.
00:26
Okay, now let's start.
00:28
So let's consider the points a and d.
00:31
Be here as the terminals that are connected to our voltage source and we will consider conventional current.
00:38
In that case, we imagine that if we take a as of the higher potential and b of lower potential, then there is a total current that comes out from the positive terminal of our battery, considering that i'm talking about conventional current.
00:56
Since there's only one path for current to flow from a to b, then logic tells us that the i total or the total current will be the same current that passes through resistor 1.
01:11
And this is also the same current that passes through resistor 2 and resistor 3.
01:19
Okay.
01:21
Knowing that this is a series circuit, which we say, which we can also consider as a voltage divider.
01:28
Okay.
01:29
Then we know that whatever is the total voltage across the circuit, which we can denote as v sub a b, it is equivalent, of course, to the sum of the individual voltages.
01:43
So it just gets allocated across each resistor.
01:47
The greater the resistance, the greater is the voltage across that element.
01:51
So we say here that v sub a b is just equivalent to voltage across resistor 1, plus the voltage across resistor 2 plus the voltage across resistor 3.
02:03
And then using omslow, we can say here that v is equal to ir.
02:08
We substitute i times r into each voltage.
02:17
So v1 becomes current i1 times r1.
02:23
Voltage 2 becomes current i sub 2 times resistance 2 and v3 becomes i3 times v3 but looking at the relationship that we have established earlier we can replace i 1 i2 and i 3 with i total because they're all equal so i instead of writing i sub 1 i'll replace it with i total r1 and this becomes i t r sub 2 plus i t r sub 3.
02:58
Just be careful in being consistent with the subscripts, because especially when you are deriving something.
03:06
So next, algebra takes over.
03:08
It dictates that we can factor out i subtotal.
03:12
So what is left in the first term is just r1.
03:16
Left in the second term is just r sub 2, and left in the third term is just r sub 3.
03:23
We still have v sub -ab in the left -hand side of our equation, dividing both sides by i total, okay, so that this is out.
03:35
Therefore, we have here, total voltage across our circuit divided by the total current.
03:43
So voltage divided by current, this actually gives us the resistance.
03:48
Therefore, the combined resistance or the equivalent resistance of these resistors in series is just the sum of the individual resistances, r sub 2 plus r sub 3.
04:00
In this case, we only have three resistors, but it actually depends on the number of resistors that are connected in series.
04:09
So let's just continue it with that.
04:12
Okay.
04:14
So this is telling us that when you combine, resistors in series, the equivalent resistance becomes greater than the individual resistances.
04:26
The more resistors you put in series, the higher the equivalent resistance becomes.
04:32
Let's proceed to the parallel circuit.
04:36
We still consider point a as having higher potential than point b.
04:41
We assume that we connect this, of course, through a voltage source.
04:46
So from positive a, conventional current i total will come out from it.
04:51
But when it reaches this junction, the current now will have more than one paths to choose from.
04:58
So some will go through this branch and that will be the current that will pass through two and three...