Look at the way that these
six and three farad capacitors are connected to each other. What's going to happen
if we hook them up to an eight volt battery? Well, like all
capacitors, charge is going to get
separated, so negatives are going to get stripped
off of the right sides of these capacitors and pulled
towards the positive terminal of the battery. But when they reach
the other side, something interesting
happens here. The charges reach this
junction, or fork in the road. And now they have a
choice in whether they're going to get deposited
onto the three farad capacitor or the
six farad capacitor. Each capacitor is going
to get some of the charge, but since the six
farad capacitor has twice the capacitance that
the three farad capacitor does, the six farad capacitor's
going to get twice as much charge stored on it as
does the three farad capacitor. So twice as many
negatives are going to get pulled off of the
right side of the six farad capacitor,
and twice as many negatives are going
to get deposited on to the left side of
the six farad capacitor. OK. So the six farad capacitor is
going to get twice as much. But exactly how much charge are
these capacitors going to get? Even though the circuit
looks a little complicated, finding the charge in this
case is actually really easy. The reason it's
going to be easy is that both of these capacitors
are hooked up directly to the terminals of the battery. In other words, the positive
side of the six farad capacitor is hooked directly up
to the positive terminal of the battery. And the negative side of
the six farad capacitor is connected directly
to the negative terminal of the battery. This means that the voltage
across the six farad capacitor is going to be the same as the
voltage of the battery, which is eight volts in this case. The same is also true for
the three farad capacitor. So the voltage across
the three farad capacitor is also eight volts. In fact, the way
these capacitors are hooked up, it's as if they
were connected to the eight volt battery all by themselves,
because they both experience the entire voltage
of the battery. Now that we know the voltage
across these capacitors, we can use the
definition of capacitance to solve for the charge. For the three
farad capacitor, we can plug-in a capacitance
of three farads and a voltage of eight volts. And we get that the
charge stored on the three farad capacitor is 24 coulombs. We could do the same
type of calculation for the six farad capacitor. We plug-in six farads
an eight volts, and we get that the charge
on the six farad capacitor is 48 coulombs. And see, just like
we said, the charge on the six farad
capacitor is twice as much as the charge on the
three farad capacitor. We call capacitors hooked
up in this way capacitors in parallel. You'll know that two capacitors
are hooked up in parallel if their positive sides
are directly connected to each other with a wire,
and their negative sides are also directly connected
to each other with a wire. We could ask ourselves
now, what should the value be of a single capacitor
whose effect on this circuit would be equivalent to that
of the individual parallel capacitors? To find the equivalent
capacitance of capacitors hooked up in parallel,
all you need to do is add up the
individual capacitances. And the reason is, just
look at these capacitors. Since their positive sides
are connected with a wire, you may as well have just
merged all of the positive sides together to form one
big positive side. And since their negative sides
are all connected with a wire, you may as well have just
merged the negative sides into one big negative side. So all you've really
done by hooking up capacitors in parallel is to
make one big capacitor out of smaller capacitors. Now, keep in mind that the
capacitance of a capacitor is proportional to the area
of the capacitor plates. So since we added the available
areas of the capacitors together to get the total
capacitance, all we need to do is to add up the
individual capacitances. Even though the charge on the
individual parallel capacitors might not be the
same, they're charge has to add up to the
total charge that would be stored on the
equivalent capacitor. So if these parallel capacitors
stored one coulomb, two coulombs, and three
coulombs individually, their equivalent capacitor
would store six coulombs. Let's try to apply these
ideas to the circuit we just examined in the
beginning of this video. The equivalent capacitance
of these six farad and three farad capacitors would be a
single nine farad capacitor. Now, let's solve for
the amount of charge that this nine farad
equivalent capacitor would store when hooked up to
the eight volt battery. Using the definition
of capacitance, we find that the charge
on a nine farad capacitor would be 72 coulombs. And this makes sense, because
remember the charge stored on the six farad
capacitor was 48 coulombs, and the charge stored on
the three farad capacitor was 24 coulombs. So the total charge on
the six farad and three farad capacitors is
72 coulombs, which is the same charge that their
equivalent capacitor stores. Let's try another problem that's
a little more challenging. Say we introduce a 27 farad
capacitor into this circuit. When the battery's
connected, the capacitors will all store charge and have
a certain voltage across them. So let's try to figure
out the charge on and the voltage across
all of these capacitors. Well, to start, we might notice
that the three farad and six farad capacitors are still in
parallel with each other, which means they have to have the
same voltage as each other. But this time, the
value of that voltage is not going to be the same
as the voltage of the battery. Because even though their
negative sides are connected directly to the negative
terminal of the battery, their positive sides are
not connected directly to the positive
terminal of the battery. This 27 farad capacitor is
getting in the way here. Similarly, the voltage
across the 27 farad capacitor is also not going to be the same
as the voltage of the battery. Because even though
it's positive side is connected directly
to the positive terminal of the battery,
it's negative side is not connected directly
to the negative terminal of the battery. So in summary, we
don't know the voltage across any of these capacitors. And if we don't know
the voltage across any of these individual
capacitors, how are we ever going to solve for
the charge on these capacitors? Well, the one thing
that we do know is that the voltage across the
whole circuit is eight volts. We just don't know the
individual voltages across each capacitor. So what we're going
to try to do is to replace these
individual capacitors with a single
equivalent capacitor. To do that, let's start
with the six farad and three farad capacitors, because we
know those are in parallel. We know they're in parallel
because their positive sides are connected
directly to each other and their negative sides
are connected directly to each other. Using the rule to combine
parallel capacitors, we get that the equivalent
capacitance of the three and six farad capacitors is a
single nine farad capacitor. So now we have a
nine farad capacitor and a 27 farad capacitor. These are connected in
series, because they're hooked up one right
after the other. Or in other words, the
positive side on one capacitor is connected to the negative
side on the other capacitor. We can replace
these two capacitors with a single
equivalent capacitor by using the formula
for adding capacitors in series, which is 1 over the
equivalent capacitance equals 1 over C1 plus 1 over C2. So plugging in the values of
nine farads and 27 farads, we get that 1 over the
equivalent capacitance equals 0.148148. Don't forget to take
1 over this number to get that the equivalent
capacitance is 6.75 farads. So we can replace the nine
farad and 27 farad capacitors with a single 6.75
farad capacitor. Now, finally, we can solve
for the charge on this 6.75 farad equivalent capacitor. Because it's positive
side is connected directly to the positive
terminal of the battery, and its negative side
is connected directly to the negative
terminal of the battery. That means the voltage across
this 6.75 farad capacitor is going to be eight volts. We can use the definition
of capacitance, and we get that the charge
on this 6.75 farad capacitor is 54 coulombs. So since this was the equivalent
capacitor for two series capacitors, both of
these series capacitors must have the same charge as
their equivalent capacitor. So both the 27 farad and
nine farad capacitors have 54 coulombs
each stored on them. At this point, we can
figure out the voltage across these two capacitors. Using the definition
of capacitance, we can plug-in 27
farads and 54 coulombs to get that the voltage
across the 27 farad capacitor is two volts. Doing the same
type of calculation for the nine farad
capacitor, we get that the voltage across the nine
farad capacitor is six volts. Notice that two
volts and six volts adds up to the voltage of
the battery, eight volts, just like they have to. And now we can find the charge
stored on the individual three farad and six farad capacitors. We know now that the voltage
across both the three farad and six farad capacitors
is going to be six volts. Because the voltage
across the individual capacitors in parallel
has to be the same as the voltage across
their equivalent capacitor. Now that we know
the voltage, we can use the definition
of capacitance. And for the three
farad capacitor, we get that the charge stored
is going to be 18 coulombs. And doing the same
type of calculation for the six farad
capacitor, we get that the charge is 36 coulombs. This makes sense, because
18 coulombs plus 36 coulombs adds up to 54 coulombs,
which was the charge stored on their equivalent
nine farad capacitor.
