CAPACITANCE
A capacitor is a device that can store electric
charge, and consists of two conducting objects
placed near each other but not touching.
Capacitors are widely used in electronic circuits.
They store charge which can later be released as
in a camera flash, and as energy backup in
computers if the power fails. Capacitors block
surges of charge and energy to protect circuits.
Very thin capacitors serve as memory for the
ones and zeroes of the binary code in the
random access memory (RAM) of computers.
• For a given capacitor, it is found that the amount of charge
Q acquired by each plate is proportional to the magnitude
of the potential difference V between them.
• Q = C V The constant of proportionality, C is called the
capacitance of the capacitor. The unit of capacitance is
coulombs per volt, and this unit is called a farad (F).
Common capacitors have capacitance in the range of 1pF (
pico farad = 10-12F ) to 103μ F (microfarad = 10-6F ). The
capacitance C does not in general depend on Q or V. Its
value depends only on the size, shape, and relative position
of the two conductors and also on the material that
separates them. For a parallel plate capacitor whose plates
have area A and are separated by a distance d of air, the
capacitance is given by
C = Ԑo A/d
• We see that C depends only on geometric factors, A and d
, and not on Q or V. The constant Ԑo is the permittivity of
free space, which has the value 8.85 X 10-12 C2/ N m2
Capacitors in Parallel
When a potential difference V is applied across
several capacitors connected in parallel, that
potential difference V is applied across each
capacitor. The total charge q stored on the
capacitors is the sum of the charges stored on all
the capacitors. Capacitors connected in parallel can
be replaced with an equivalent capacitor that has
the same total charge q and the same potential
difference V a the actual capacitors.
q1 = C1V
q2 = C2V q3V
Q = q1 + q2 + q3 = Ceq V
C1V + C2V + C3V = CeqV
Ceq = C1 + C2 + C3
When a potential difference V is applied across
several capacitors connected in series, the
capacitors have identical charge q. The sum of
the potential differences across all the
capacitors is equal to the applied potential
difference V. Capacitors that are connected in
series can be replaced with an equivalent
capacitor that has the same charge q and the
same total potential difference V as the actual
series capacitance
q
V V1 V2 V3
Ceq
q q q
q
C1 C2 C3 Ceq
1
1 1 1
Ceq C1 C2 C3
Example 1. Capacitor calculations.
a) Calculate the capacitance of a parallel plate
capacitor whose plates are 20 cm X 3 cm and
are separated by a 1 mm air gap.
b) What is the charge on each plate if a 12V
battery is connected across the two plates?
C) What is the electric field between the plates?
d) Estimate the area of the plates needed to
achieve a capacitance of 1F, given the air gap d.
Solution: a) The area A = (20 X 10-2 m) ( 3 X 10-2
m ) = 6 X 10-3 m2
The capacitance is then:
C = Ԑo A/d = 8.85 X 10-12 C2/ N m2 ( 6X10-3 m2 /
1 X 10-3 m = 53 Pf
b) The charge on each plate is: Q = CV = ( 53 X 1012F) ( 12V) = 6.4 X 10-10 C
c) E = V / d = 12V / 1 X 10-3 m = 1.2 X 104 V /m
d) A = Cd / Ԑo = ( 1F ) ( 1 X 10-3 m ) / 9 X 10-12 C2
/ N m2 ) = 108 m2
• DIELECTRICS
In most capacitors there is an insulating sheet of
material, such as a paper or plastic called a dielectric
between the plates. This serves several purposes. First,
dielectrics do not break down as readily as air, so higher
voltages can be applied without charge passing across the
gap. Furthermore, dielectric allows the plates to be
placed closer together without touching, thus allowing an
increased capacitance because d is less. Thirdly, it is
found experimentally that if the dielectric fills the space
between the two conductors, it increases the capacitance
by a factor K, known as the dielectric constant. Thus for a
parallel-plate capacitor:
C = K Ԑo (A/d) this can also be written
as C = Ԑ A / d where Ԑ K Ԑo is called the
permeability of the material.
The potential difference between the
plates, Vo is given by Q = Co Vo
Storage of Electrical Energy
A charged capacitor stores electrical
energy by separating positive and negative
charges. The energy stored in a capacitor will be
equal to the work done to charge it. The net
effect of charging a capacitor is to remove
charge from one plate and add it to the other
plate. This is what the battery does when it is
connected to a capacitor. Initially, when the
capacitor is uncharged, no work is required to
move the first bit of charge over.
As more charge is transferred, work is needed to
move charge against increasing voltage V. The work
needed to add a small amount of charge ∆q, when
a potential difference V is across the plates is ∆W
= - V ∆q. The total work needed to move total
charge Q is equivalent to moving all the charge Q
across a voltage equal to the average voltage during
the process. The average voltage is
( Vf 0 ) / 2 = Vf / 2 where Vf is the final voltage, so the
work to move the total voltage Q from one plate
to the other is: W = Q Vf / 2
Thus we can say that the electric potential energy,
PE, stored in a capacitor is PE = energy = ½ Q Vf
where V is the potential difference between the
plates and Q is the charge on each plate. Since Q
= CV we can also write
PE = ½ QV =
½ CV2 = ½ Q2 / C
Example 1. Energy Stored in Capacitor. A camera
flash unit stores energy in a 150μF capacitor at 200
V. How much electric energy can be stored?
Solution: PE = ½ CV2
= ½ ( 150 X 10-6F) ( 200 V )2 = 3 J
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