### Video Transcript

A potential divider has an input
voltage of 48 volts. The resistance of the second
resistor, π
two, is 100 kiloohms. The output voltage is drawn across
the second resistor, π
two. What resistance must the first
resistor π
one have in order to produce an output voltage of 32 volts?

Okay, so in this case, weβre
dealing with a potential divider circuit. So letβs first start by drawing our
potential divider circuit. Letβs first start by recalling that
a potential divider circuit consists of two resistors connected in series. Weβll call the resistance of our
first resistor π
one and the resistance of our second resistor π
two, as weβve
been told to in the question. We also recall that across this
part of the circuit, some sort of power source is connected, often a battery, but
not necessarily so. And this power source is what
provides whatβs known as the input voltage. Letβs call this input voltage π
subscript in. And in this case, in the question,
weβve been told that this input voltage is equal to 48 volts.

Now, additionally, weβve been told
that the resistance of the second resistor π
two is 100 kiloohms. Thatβs 100000 ohms because the
prefix kilo- means 1000. Weβve also been told that the
output voltage in our potential divider circuit is drawn across the second
resistor. What this essentially means is that
we connect a pair of wires here and here so that we can connect some components in
parallel with our resistor π
two. In this case, weβre not really
worried about which components are connected here, but just that the output voltage,
which we will call π subscript out, is drawn across our second resistor.

Now, the reason that we know that
this is the output voltage is because the voltage across our second resistor must be
the same as π out. The reason for this is that our
resistor π
two and whatever components are connected here are connected in
parallel. And components in parallel have the
same potential difference across them. Therefore, to reiterate, the
potential difference across resistor π
two is π subscript out. Thatβs the output voltage. And thatβs also the potential
difference across whatever components are connected here and is therefore the output
voltage.

Weβve been told in the question
that the output voltage must be equal to 32 volts. And weβve been asked to try and
work out what the resistance of the first resistor π
one must be in order for this
to be true. So in order to answer this
question, we will need to recall the potential divider equation. This equation tells us that the
potential difference across resistor π
two, which weβre calling π two, is equal to
the resistance π
two divided by the sum of the resistances π
one plus π
two all
multiplied by the input potential difference π subscript in.

Now remember, the potential
difference across our second resistor π two is the same as our output potential
difference because theyβre connected in parallel. So we can replace π two with π
out here, at which point we see weβve already got a value for π out, a value for π
two, and a value for π in. We just need to rearrange to solve
for π
one. We can do this by multiplying both
sides of the equation by π
one plus π
two and dividing both sides of the equation
by π out. Because this way on the left-hand
side weβve got π out over π out, which is equal to one. And on the right-hand side, weβve
got π
one plus π
two both in the numerator and denominator.

Then we simply subtract π
two from
both sides of the equation so that weβre left with π
one on the left, at which
point we simply substitute in the values on the right-hand side, taking care to
notice that in this fraction the unit of volts is both in the numerator and
denominator and that if weβre going to stick with kiloohms, then our final answer
for π
one is going to be in kiloohms as well. When we simplify all of the
right-hand side, we find that our final answer is π
one is equal to 50
kiloohms.