Week 4: Capacitance 141
have been any number! We just add the potentials across as many as we have (with the same charge
on each capacitor) to get the total potential difference for the series row.
These two forms must beequalfor equalQon the two arrangements. That’s thedefinitionof
the total capacitance of the upper arrangement – the equivalentsingle capacitor one could replace
the row with and get the same potential difference for the givenQ. Equating them and cancelling
the commonQ, we get:
1
Ctot=
1
C 1
+
1
C 2
+
1
C 3
+...=
∑
i1
Ci(236)
where again the ... and final summation indicates that we just sum over as many capacitors as there
are in the series row. For capacitors in series, thereciprocalof the total capacitance equals the sum
of thereciprocalsof the individual capacitors in series.
Why is this rule so odd? Because in series, we would get a more intuitive result by thinking of
adding capacitors as if they werevolticitors, and “volticitance” is the reciprocal of the capacitance!
Why is series addition of capacitors important and useful? Putting capacitors in seriesreduces
the total capacitance (check this for yourself!) and isn’t a big capacitor better than a small one?
Well, yes and no. It turns out that most capacitors can only support afinite voltageacross them
beforedielectric breakdownoccurs across the intervening gap, shorting them out and burningthem
out. If you want to put more voltage than that maximum across a capacitor in a circuit (and don’t
have any rated at the desired voltage) you can put a bunch of capacitors rated at a lower voltage
in series until youcanput the desired voltage across them without exceeding the maximumfor any
single capacitor in the series leg. Or, you might have a bunch of big capacitors in your box and need
a smaller one that wasn’t in your box – adding several up in series can let you save a trip to radio
shack!
So how about parallel? When several circuit elements are connectedon both sides by a common(^1) C 1 C 2 C 3
Q Q 2 Q 3
Ctot
Qtot
V
V
Figure 43: Find the total capacitance of a much of capacitorsin parallel.
conductor, the conductor on each side isequipotential. That means that all of the elements have the
same potential differenceacross them. Note that this time I am not bothering to explicitly indicate
the charge−Q 1 etc on the other plate of each capacitor. Recall, a capacitor is presumed toalways
have equal and opposite charges on its plates unless someone goesfar out of their way to make up
a problem with something different.
In figure 43eachcapacitor in the top arrangement has a potentialV across it. Therefore the
first capacitor has a chargeQ 1 =C 1 V, the second has a chargeQ 2 =C 2 V, the thirdQ 3 =C 3 V.
The equivalenttotalcapacitanceCtotwith thesamevoltageV across it has a chargeQtot=CtotV
on it. For them to be the same, the total charge store on the top arrangement has to equal that on
the bottom.