Capacitors in Parallel Video Lecture - Physics Class 12 - NEET

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.