Video Transcript
The circuit in the diagram contains
two capacitors connected in series. What is the total capacitance of
the circuit? Answer to the nearest
microfarad.
Okay, so in this question, we’ve
been given a diagram that shows a simple circuit where we’ve got a cell connected in
series with two capacitors. The capacitances of each of these
two capacitors are labeled on the diagram in units of microfarads. This one on the left has a
capacitance of 150 microfarads, while this one has a capacitance of 250
microfarads. We’re being asked to find the total
capacitance. To do this, we need to recall that
when we have several capacitors connected in series, then one over the total
capacitance, which we’ve labeled as 𝐶 subscript T, is given by the sum of the
reciprocals of the individual capacitances. So if we label those capacitances
with subscripts one, two, three, etcetera, then that’s one over 𝐶 one plus one over
𝐶 two plus one over 𝐶 three and so on for any further capacitors connected in
series.
In this question, we’ve only got
two capacitors in series in our circuit, and so we only want the first two terms on
the right-hand side of this equation. Let’s label this 150-microfarad
capacitor as 𝐶 one and this 250-microfarad one as 𝐶 two. Then in this equation, in place of
the quantities 𝐶 one and 𝐶 two, we can use our values of 150 and 250
microfarads. This gives us an expression which
says that one over the total capacitance 𝐶 subscript T is equal to one over 150
microfarads plus one over 250 microfarads.
In order to add together two
fractions, we want those fractions to have a common denominator. One simple way to do this is to
cross multiply. So we take the 250 from the
denominator of the right-hand fraction, and we multiply the numerator and
denominator of the left-hand fraction by this 250. Then we take the 150 from the
denominator of the left-hand fraction, and we multiply the numerator and denominator
of the right-hand fraction by this 150. We’ve now got two fractions which
both have the same denominator equal to 250 multiplied by 150 in units of
microfarads. This works out as 37500
microfarads. We can now add together these two
fractions to get a result of 400 divided by 37500 microfarads. Evaluating this fraction then gives
a result of 0.0106 recurring with units of inverse microfarads.
Notice that we got these units of
inverse microfarads because we had a quantity with units of microfarads in the
denominator of the fraction. Now having units of inverse
microfarads, or one divided by microfarads, makes complete sense here because what
we calculated was one divided by the total capacitance 𝐶 subscript T. In the question, we’re trying to
find the value of the total capacitance itself, so that’s the value of 𝐶 subscript
T. That means that we need to take the
reciprocal of this result for one over 𝐶 subscript T in order to get our
answer. So we then have this expression
here for the total capacitance 𝐶 subscript T. One divided by a quantity in units
of one over microfarads will produce a result with units of microfarads.
When we evaluate the expression, we
find that this result is equal to exactly 93.75 microfarads. This value is equal to the total
capacitance of the circuit, which is what we were asked to find. But the last thing left to do is to
notice that we were asked to give this answer to the nearest microfarad. Rounding this result to the nearest
microfarad gives us our answer for the total capacitance as 94 microfarads.
