Capacitance

1
Capacitance
When a potential difference, also called
voltage, is applied to a pair of parallel
conductive plates closely spaced, conduction
electrons are moved from the plate connected to
positive side of the battery to the other plate
connected to the negative side. The electrons
will continue to move until the voltage across
the plates equals that of the battery. If the
battery is disconnected from the plates, the
electrostatic attractive force between the
charged plates will keep the charges on the inner
surfaces and the charges will remained stored in
the plates until conductive path is made between
the plates (short) .
Q
-Q
Switch
Battery
2
Capacitance
Units of C Coulomb/Volt Farads or F Analog to
water storage by a reservoir
Electric vs
Water Reservoir
Q
Water Volume V

depth of Water C
Area of the
reservoir QCV
volume area x depth
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Capacitance Example
4
Capacitance Example
5
Cylindrical Capacitor
In the real world, the plates of a capacitor are
typically not flat, but are curved such as two
concentric conducting cylinders as in the below
example
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Cylindrical Capacitor
7
Cylindrical Capacitor
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Spherical Capacitor
In this case we consider the case of two
concentric conducting spheres as a capacitor
Recall that the charge on the inner conductor can
be treated as if the charged is concentrated at
the center and field lines terminate at inner
surface of the outer conductor. The field between
the two conductors is
Therefore voltage between the spheres
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Spherical Capacitor
We can then rewrite the voltage as
Therefore the capacitance of spherical capacitor
We can determine the capacitance of a single
isolated sphere by assuming the missing plate is
a conducting sphere of infinite radius , b ?
and substitute R a
As previously mentioned, the farad is a huge
unit. So how big does an isolated sphere must be
have a capacitance of 1F.
R 8.99x109 meters
The distance from Earth to the moon is only
3.84x108m !!
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Returning to Parallel plates
Recalling, the capacitance of parallel plates in
which plates of area A and separated by distance
d was determined to be
Example Find C for a parallel plate capacitor of
A2cm2, and d1mm Using MKS units, we have A
2x10-4m2, d 10-3m, C 8.85x10-12 C2/nM2 x
2x10-4m2 / 10-3m 1.77x 10-12 F 1.77??F The
Farad is indeed a very big unit
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Summary of Different Capacitor Geometry
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Capacitor Combination
When a combination of capacitors exist in a
circuit, an equivalent capacitor can usually
replace that combination and therefore simplify
the circuit.
Note When a voltage is applied across several
capacitors connected in parallel, voltage is
applied across each capacitor. The total charge
stored on the capacitors is the sum of charges
stored on each capacitor.
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Capacitor Combination
14
Capacitor Combination
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Capacitor Combination
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Energy Stored in a
As we charge a capacitor from q0 to the final q
CV, the first few arriving electrons encounter
effectively very little electric field therefore
very little work was required to perform this
initial charge transfer from one plate to the
other. However, as charge accumulates,
increasingly larger amount of work is needed to
continue the transfer. At a given instant, a
charge q has been transferred from one plate to
the other. The voltage V between the plates at
that instant will be q/C. If extra increment of
charge is then transferred, the increment of work
required is
The total work for the total charging process to
final q
Which is stored as potential energy
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Dielectrics in Capacitor
A dielectric is an isolator such as glass or
paper. Dielectric materials in capacitor decrease
the electric field and increase the capacitance.
Ef Ei/? ? dielectric constant
i

• We also have
• Vf Vi/?, Qf Qi (charge is constant)
• Cf Q/V Qi/(Vi/?) ?Ci

f

• Cf ?Ci

We can increase the capacitance by filling the
capacitor with material having a large ?
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Dielectrics in Capacitor
The dielectric constant ? for different materials
has broad spectrum of values as shown in the
table The product ? ?0 ? Is called the
permittivity of the dielectric . For a parallel
capacitor filled with dielectric C ? ?0 A/d
? A/d