Capacitance
and Dielectrics
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Definition of Capacitance
• A capacitor consists of two
conductors separated by an
insulator. The capacitance of a given
capacitor depends on its geometry
and on the material (called a
dielectric) that separates the
conductors.
• Capacitance Is a capacity
• The capacitance C of a capacitor is
defined as the ratio of the magnitude
of the charge on either conductor to
the magnitude of the potential
difference between the conductors:
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• The SI unit of capacitance is the farad
(F), which was named in honor of
Michael Faraday:
• A parallel-plate capacitor consists of two
parallel conducting plates, each of area
A, separated by a distance d. When the
capacitor is charged by connecting the
plates to the terminals of a battery, the
plates carry equal amounts of charge.
One plate carries positive charge, and
the other carries negative charge.
• Suppose that we have a capacitor rated
at 4 pF. This rating means that the
capacitor can store 4 pC of charge for
each volt of potential difference between
the two conductors.
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Calculating Capacitance
• We can derive an expression for the capacitance of a pair of
oppositely charged conductors in the following manner:
– assume a charge of magnitude Q,
– calculate the potential difference using the techniques described
in the preceding chapter.
– We then use the expression C = Q/∆V to evaluate the
capacitance.
• The capacitance of a pair of conductors depends on the geometry of
the conductors.
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Parallel-Plate Capacitors
• the capacitance of a parallel-plate capacitor is proportional to the
area of its plates and inversely proportional to the plate separation
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A circuit consisting of
a capacitor, a battery, and a switch.
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Cylindrical Capacitors
• A cylindrical capacitor consists of a solid cylindrical conductor of
radius a and length l surrounded by a coaxial cylindrical shell of
radius b.
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• End view. The electric field lines are radial.
The dashed line represents the end of
the cylindrical gaussian surface of
radius r and length l.
• Using Gauss’s law that the magnitude of the electric field of a
cylindrical charge distribution having linear charge density λ is
E = 2ke λ/r
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Combinations of Capacitors
• Parallel Combination
• the individual potential differences across capacitors connected in
parallel are the same and are equal to the potential difference
applied across the combination.
• the total charge on capacitors connected in parallel is the sum of the
charges on the individual capacitors.
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• the equivalent capacitance of a parallel combination of capacitors is
the algebraic sum of the individual capacitances and is greater than
any of the individual capacitances.
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• Series Combination
• the charges on capacitors connected in series are the same.
• the total potential difference across any number of capacitors connected
in series is the sum of the potential differences across the individual
capacitors.
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• the inverse of the equivalent capacitance is the algebraic sum of the
inverses of the individual capacitances and the equivalent
capacitance of a series combination is always less than any
individual capacitance in the combination.
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Energy Stored in a Charged Capacitor
• A plot of potential difference versus charge for a capacitor is a
straight line having a slope 1/C. The work required to move charge
dq through the potential difference ∆V existing at the time across the
capacitor plates is given approximately by the area of the shaded
rectangle. The total work required to charge the capacitor to a final
charge Q is the triangular area under the straight line,
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Capacitors with Dielectrics
• A dielectric is a nonconducting material, such as rubber, glass, or
waxed paper. When a dielectric is inserted between the plates of a
capacitor, the capacitance increases.
• A charged capacitor before and after insertion of a dielectric between
the plates. The charge on the plates remains unchanged, but the
potential difference decreases from ∆V0 to ∆V = ∆V0/κ. Thus, the
capacitance increases from C0 to κC0.
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• the capacitance increases by the factor κ when the dielectric
completely fills the region between the plates.
• For a parallel-plate capacitor, we can express the capacitance when
the capacitor is filled with a dielectric as
• a dielectric provides the following advantages:
– Increase in capacitance
– Increase in maximum operating voltage
– Possible mechanical support between the plates, which allows the plates to be
close together without touching, thereby decreasing d and increasing C.
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Types of Capacitors
• Three commercial capacitor designs.
– A tubular capacitor, whose plates are separated by paper and then rolled into a
cylinder.
– A high-voltage capacitor consisting of many parallel plates separated by
insulating oil.
– An electrolytic capacitor.
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Quick quiz
• If you have ever tried to hang a picture or a mirror, you know it can
be difficult to locate a wooden stud in which to anchor your nail or
screw. A carpenter’s stud-finder is basically a capacitor with its plates
arranged side by side instead of facing one another, as shown in
figure. When the device is moved over a stud, does the capacitance
increase or decrease?
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PR
• Sebuah kapasitor keping-paralel diisi dengan dua buah
dielektrik seperti telihat pada gambar. Tunjukkan bahwa
– Sistem ini dapat dipandang sebagai dua kapasitor seluas A/2
yang terhubung secara paralel dan
– Kapasitansinya naik sebesar faktor (κ1+ κ2)/2.
A
d
κ1
κ2
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