Transcript NASC 1110 - 03 - The University of Toledo

Lecture 4
Capacitance and Capacitors
Chapter 16.6  16.10
Outline
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Definition of Capacitance
Simple Capacitors
Combinations of Capacitors
Capacitors with Dielectrics
Capacitance
History
Introduction
General definition
The capacitance C of a capacitor is the ratio of the
charge (Q) on either conductor plate to the
potential difference (V) between the plates.
Q
C
V
Units of capacitance are farads (F)
1F  1C/1V
C(Earth) ~ 1 F
C(adult) ~ 150 pF = 150 1012 F
The Parallel-Plate Capacitor
The capacitance of a parallel-plate capacitor
whose plates are separated by air is:
A
C = є0 
d
A is the area of one of the plates
d is the distance between the plates
є0 is permittivity of free space
More about capacitors
Capacitors
Problem: A parallel-plane capacitor has an area of
A=5cm2 and a plate separation of d=5mm. Find its
capacitance.
Unit conversion: A = 5 cm2 = 5 104 m2
d = 5 mm = 5 103 m
C = є0 A/d = 8.85 1012 C2 / (N m2) 5 104 m2 / 5 103 m
= 8.85 1013 C2/(N m)= 8.85 1013 F = 0.885 pF
N/C = V/m  C/N = m/V, F=C/V
C2/(N m)=C (C/N)/m = C (m/V)/m = C/V = F
Combinations of Capacitors
In real electric circuits capacitors can be connected
in various ways.
In order to design a circuit with desired capacitance,
equivalent capacitance of certain combinations of
capacitors can be calculated.
There are 2 typical combinations of capacitors:
• Parallel combination
• Series combination
Parallel Combination
Parallel Combination
• The left plate of each capacitor is connected to the
positive terminal of a battery by a wire 
• the left plates are at the same potential 
• the potential differences across the capacitors are
the same, equal to the voltage of the battery (V).
• The charge flow ceases when the voltage across
the capacitors equals to that of the battery and the
capacitors reach their maximum charge.
Examples
Q = Q 1 + Q2
Q = Ceq V
Q1 = C1 V
Ceq V = C1 V + C2 V
Ceq = C1 + C2
Q2 = C2 V
Series Combination
Series Combination
The magnitude of the charge is the same on all the
plates.
The equivalent capacitor must have a charge –Q on
the right plate and +Q on the left plate.
Q
V = 
Ceq
V = V1 + V2
V1 = Q/C1
V2 = Q/C2
Examples
Q Q Q
=+
Ceq C1 C2
1
1 1
=+
Ceq C1 C2
Energy Stored in a Capacitor
The work required to move a charge Q through a
potential difference V is W = V Q.
V = Q/C, Q is the total charge on the capacitor.
The voltage on the capacitor linearly increases with
the magnitude of the charge.
Additional work increases the energy stored.
W = ½ Q V = ½ (C V) V = ½C (V)2 = Q2/2C
Capacitors with Dielectrics
A dielectric is an insulating material.
The dielectric filling the space between the plates
completely increases the capacitance by the factor
 > 1, called the dielectric constant.
If V0 is the potential difference (voltage) across a
capacitor of a capacitance C0 and a charge Q0 in
the absence of a dielectric.
Filling the capacitor with a dielectric reduces the
voltage by the factor  to V, so that V = V0/.
C = Q0/V = Q0/V0/ =  Q0/V0 =  C0
Dielectric Strength
For a parallel-plate capacitor: C =  є0 A/d
The formula shows that the capacitance can be
made very large by decreasing the plate separation.
In practice, the lowest value of d is limited by the
electric discharge through the dielectric.
The discharge occurs when the electric field in the
dielectric material reaches its maximum, called
dielectric strength.
Dielectric strength of air is 3 106 V/m.
Summary
• Capacitance is defined as the charge over the
potential difference
• Capacitance of parallel-plate capacitor is directly
proportional to the plate area and inversely
proportional to the plate separation
• The equivalent capacitance of a parallel
combination of capacitors equals to the sum of
individual capacitances
• The inverse equivalent capacitance of a series
combination of capacitors equals to the sum of the
inverse individual capacitances
• Placing a dielectric between the plates of a
capacitor increases the capacitance by a factor ,
called the dielectric constant