Lesson #7
Capacitors and Dielectrics I
I.
Capacitor
A.
A capacitor is an electrical device that stores energy in an electric field created by
the separation of charge between two conductors.
+
+
+
+
+
B.
r
E
Electric
Piggy Bank
V
Definition of Capacitance
The capacitance of a capacitor is defined as the ratio of the charge transferred
to one of the conductors of the capacitor divided by the electric potential
between the conductors.
When thinking about capacitance, you should remember the phrase “charge
CAPACITY”!!!
C.
Units of Capacitance -
D.
Charging a Capacitor
Two neutral plates are connected to a battery to form the electrical circuit shown
below:
+ Battery -
Switch
Since there is no net charge on either plate, there is no electric field between the
plates and no electric potential energy stored.
A battery can be considered to be a charge pump. When the switch is closed,
the battery will pump draw positive charge into the negative terminal and pump
the positive charge out of the positive terminal. This is analogous to the flow of
water using a water pump.
Thus, an amount of charge +Q leaves the right plate traveling clockwise in the
circuit and is placed on the left plate as shown below:
+Q
-Q
Direction of
Charge Flow
+ Battery -
Switch
r
The separation of charge produces an electric field E between the plates. Thus,
an electric potential difference exists across the plate and electric potential energy
is stored in the system equal to work done by the battery in transferring the charge
between the plates.
EXAMPLE: A battery transfers 2.00 µC of charge between the plates creating an
electric potential difference of 4.00 v across the capacitor. What is the capacitance of the
capacitor?
E.
Factors Influencing Capacitance
The amount of charge that can be stored on a capacitor for a given electric
potential difference (i.e. capacitance) depends on
1) the geometry of the capacitor
2) the material placed between the conductors of the capacitor.
1.
Geometry Effects (Water Analogy)
To help you understand the importance of geometry in a capacitor, consider the
following water analogy involving gravitational potential instead of electrical
potential.
h
In this analogy, the charge stored on a capacitor’s conductor corresponds to the
amount of water stored in a glass. The gravitational potential of the water is
related to the height of the water and is the analog of the electric potential across
the capacitor.
In our diagram, it is easy to see that the wider glass (largest area) contains more
water when the water in both glasses is at the same height.
By analogy, a capacitor whose conductor has a larger surface area can hold
more charge for a given electric potential difference. Put another way, it takes
more charge to achieve the same electric potential difference if for a capacitor
whose conductor has a larger surface area.
Self Study Problem: Consider Gauss’Law and the relationship between the
electric field and electric potential difference to see why increasing the area of the
conductor would decrease the electric potential and hence increase a capacitor’s
capacitance. (Hint: Choose a simple geometry for your capacitor’s conductor.)
The water analogy can be useful in getting a handle on many concepts
involving capacitors.
For instance, we could consider the problem of placing the same amount of
charge upon two capacitors of different capacitance. The water analog of our
problem is shown below:
In the diagram, the water reaches a greater height in the glass with the lower water
capacity. Thus, the smaller capacitor (lowest capacitance) will be charged to a
higher voltage (electrical potential)!!
2.
Material Effects (Dielectrics)
In the diagram below, a charge of Q is transferred between the plates of a
r
capacitor setting up an electric field, E , and electric potential, ∆V, between the
i
plates.
∆V
-Q
+Q
r
E
i
We now place a dielectric slab (e.g. glass, paper, mica, oil, etc.) between the
plates as shown below:
-+
+Q
-Q
-+
-+
The initial electric field causes the dielectric to be _______________________.
This produces an internal _________________
__________________ that
______________________ the initial electric field. Thus, the resultant electric
field is _______________________ than the initial electric field.
Since the total electric field is ____________, the electrical potential difference
across the plates of the capacitor is also _________________ since
Br r
? V = − ∫E •d s .
A
Thus, more _______________ can be added to the capacitor to ______________
the electrical potential difference back to its original value.
Therefore, the capacitor’s capacitance has ______________________________.
If this section above, we saw that placing a dielectric material between the
conductors of a capacitor changes the electric field between the plates.
Self Study: Doesn’t our results violate Gauss’Law since the net enclosed charge
didn’t change!!
II.
Calculating the Capacitance of a Capacitor
Because the capacitance of a capacitor depends on how it is constructed, the re is
no universal formula for determining the capacitance of a capacitor given the
device’s dimensions and dielectric. Instead, you must use definition of
capacitance and the following procedure to derive the necessary formula.
Need Formula for
Determining Capacitance
Of a Given Capacitor
Transfer a Charge of +Q
Between the Conductors of
The Capacitor
Determine the Resulting
Electrical Potential Difference
Between the Conductors
Use the Definition of
Capacitance, C = Q/V, to
Calculate the Capacitance
EXAMPLE: Determine the capacitance of the parallel-plate capacitor whose plates are
separated 0.45 mm and have a plate area of 256 cm2.
SOLUTION:
STEP 1: Transfer a charge of +Q between the conductors
d
y
+Q
r
E
-Q
Plate Area - A
STEP 2: Find the electric potential difference across the plates
Using Gauss’Law, the electric field between the plates is
We now calculate the voltage across the plates as
x
STEP 3: Use the definition of capacitance
C ≡Q =
V
C=
This formula is only good for a parallel-plate capacitor with no dielectric between the
plates!!!
III.
Energy Stored in a Capacitor
As we transfer positive charge on to the positive plate of the conductor, we must
do work against the net repulsive force of the other positive charges that already
exist on the conductor. Thus, we are storing electrical potential energy as
discussed in the proceeding lesson.
The electrical potential energy stored in a capacitor of capacitance C when
charged to an electric potential difference of V is given by
PROOF #1: Calculus
Consider the electrical potential energy stored as an incremental piece of charge is
moved from the negative to the positive plate as shown below:
dq
-
+
V
dU =
But, at every instant of time, we have that
Thus, using Calculus we have the relationship
Therefore, the electrical potential energy stored in moving the infinitesimal
charge dq is given by
We then integrate to find the total energy stored in charging up the capacitor.
PROOF 2: Geometric Proof (non-calculus)
The electrical potential energy stored in a capacitor is the area under a VoltageCharge graph.
V
Q
The first piece of positive charge cross the gap when the charge on the plate, Q, is
__________________ and an electric potential difference of _______________.
The last piece of positive charge crosses the gap with the plate fully charged to Q
so the electrical potential difference is ________________________.
From the definition of capacitance, we know that the V-Q graph for a capacitor is
linear with slope of 1/C. Drawing the line between our two points, we have that
U = Area =
U=
Example: How much energy is stored in a 100 µf capacitor at 100 volts?
IV.
Parallel and Series Electric Circuits
A.
Parallel –
Two electrical circuit elements (example: two capacitors) are in parallel if the
both have the SAME ELECTRICAL POTENTIAL DIFFERENCE.
IMPORTANT: By the “same voltage” we mean that you must be able to
simultaneously measure the voltage drop across both elements in a single
measurement and not that the numerical values for the voltage drops are the
same!!!
Example: If we place our voltmeter leads between points A and B, we will
simultaneously measure the voltage drops across both capacitor 1 and capacitor 2. Thus,
these capacitors are in parallel.
C1
B
A
C2
Example 2: In the drawing below, are the three capacitors in parallel?
B
+ 3V -
A
+ 3V -
C
+ 3V -
B.
Series –
Two electrical circuit elements are in series if the individual charges leaving the
first circuit element must enter the second element.
We will see later that this definition is the same as saying the same current flows
through both elements. Thus, we must be able to simultaneously measure the
current in both elements with an ammeter.
Example: The two capacitors below are in series.
C
B
A
C1
C2
C.
Neither Parallel nor Series –
Many students assume that all circuit elements are either in parallel or series.
This is false as demonstrated in the figure below.
C1
C2
C3
C4
C5
C6
C7
The capacitors C1 and C7 are in series, but no other capacitors are in series or
parallel!! The circuit composed of capacitors C2, C3, C4, C5, and C6 is called a
bridge circuit and is very useful. Bridge circuits using resistors, transducers, and
other circuit elements are often used in designing instrumentation amplifiers for
medical applications.
Question: Why are parallel and series circuits so important?
Answer: Because we can combine elements in series or parallel to create a single
equivalent circuit element. This simplifies the mathematical analysis of the
circuit. Furthermore, any group of elements that are not in either parallel or series
can always be converted to new elements that are in series and/or parallel.
V.
Capacitors in Series and Parallel
A.
Series - A group of capacitors in series are equivalent to a single _____________
capacitor determined by the formula:
C1
C2
Ceq
C3
≡
Note: Putting capacitor’s in series is like making the space between the plates
_______________________ which makes the capacitance ___________________.
B.
Parallel - A group of capacitors in parallel are equivalent to a single
________________ capacitor determined by the formula:
C1
C2
C3
≡
Ceq
Note: Placing the capacitors in parallel is like ______________________ the
plate area which ______________________ the capacitance.
C.
Special Case – Two Capacitors in Series
The following formula is very convenient for simplifying circuits.
C1
Ceq
C2
≡
D.
Special Case – “N” Equal Capacitor’s in Parallel
C
Ceq
C
≡
EXAMPLE: What is the equivalent capacitance of the following circuit?
C
A
3µf
2µf
B
1µf
D
3µf
V.
Capacitors and Dielectrics