Capacitance and Dielectrics
You will learn:
The nature of capacitors, and their ability to store charges
Capacitors connection in a networks
Calculate the amount of energy stored in capacitors
3.1 Capacitors and Capacitance:
• A capacitor is a device which stores electric charges. Capacitors mostly consist
of two conductors with an insulating medium placed between them (figure 3.1).
• The insulating medium can be either vacuum or an insulator material.
figure 3.1 Capacitor structure
• Capacitors vary in shape and size, but the basic configuration is two conductors
carrying equal but opposite charges.
• There are some common types of capacitors, parallel plate capacitors, cylindrical
capacitors, and concentric capacitors. In this chapter, parallel plate capacitors will
• In circuit diagrams, a capacitor is represented by either of below symbols:
Figure 3.2 Capacitor symbols (straight or curved)
• In most practical applications, each conductor initially has zero net charge (Q) and electrons
are transferred from the voltage source to the conductors; this is called charging the
capacitor. Then the two conductors have charges with equal magnitude and opposite sign,
and the net charge on the capacitor as a whole remains zero.
• A potential difference Δ𝑉 is created with the positively charged conductor at a higher
potential than the negatively charged conductor.
• The amount of charge Q stored in a capacitor is linearly proportional to ΔV.
• To charge a capacitor, you need a power supply source or a battery.
• Once the charges Q and – Q are established on the conductors, the battery is disconnected.
This gives a fixed potential difference between the conductors (that is, the potential of the
positively charged conductor with respect to the negatively charged conductor ) that is just
equal to the voltage of the battery.
• Each capacitor has capacitance. So, capacitance is a measure of the capacity of
storing electric charges for a given potential difference. The SI unit of capacitance
is the farad.
• 1 Farad (F) = 1 coulomb/volt
• Farad is a large quantity, A typical capacitance is in the picofarad (pF) to millifarad
range (mF), microfarad (μF) or nanofarad (nF).
• Mathematically, we can say
• The electric field at any point in the region between the conductors is proportional to the
magnitude Q of charge on each conductor. It follows that the potential difference Δ𝑉 (or
written Vab) between the conductors is also proportional to Q.
• Note: Don’t confuse the symbol for capacitance C with the abbreviation of C for
• If we double the magnitude of charge on each conductor, the charge density at each
point doubles, the electric field at each point doubles, and the potential difference
between conductors doubles; however, the ratio of charge to potential difference does
• The capacitance of the parallel plate capacitor:
Depends on the medium between the two plates.
Is directly proportional to the area of the plated.
Is inversely proportional to the plate separation.
A is the area of plates,
d is the distance between the two plates and
𝜀0 is the permittivity of a vacuum (free space).
Figure 3.3 A charged parallel-plate capacitor.
A commercial capacitor is labelled
with the value of its capacitance. For these
capacitors, C=2200, 1000 and 470 μF.
An assortment of commercially
3.2 Connection of capacitors in Series and Parallel
3.2.1 Capacitors in series:
• Suppose two initially uncharged capacitors are connected in series, as shown in
Figure 3.4. A potential difference is then applied across both capacitors. The left
plate of capacitor 1 is connected to the positive terminal of the battery and becomes
positively charged with a charge +Q, while the right plate of capacitor 2 is
connected to the negative terminal and becomes negatively charged with charge –Q
as electrons flow in.
• What about the inner plates? They were initially uncharged; now the outside plates
each attract an equal and opposite charge. So the right plate of capacitor 1 will
acquire a charge –Q and the left plate of capacitor +Q.
Figure 3.4 Capacitors in series and an equivalent capacitor 8
The potential differences across capacitors C1 and C2 are
The total potential difference is simply the sum of the two individual potential
The generalization to any number of capacitors connected in series is
Note: The term of (total) is used instead of (equivalent) sometimes to show a total
values of something.
3.2.2 Capacitors in Parallel:
• Suppose we have two capacitors C1 with charge Q1 and C2 with charge Q2
which are connected in parallel, as shown in Figure 3.5. The left plates of
both capacitors C1 and C2 are connected to the positive terminal of the
battery and have the same electric potential as the positive terminal.
Similarly, both right plates are negatively charged and have the same
potential as the negative terminal.
Figure 3.5 A parallel connection of two capacitors.
(b) The equivalent single capacitor
These two capacitors can be replaced by a single equivalent capacitor Ceq with a
total charge Q supplied by the battery. However, since Q is shared by the two
capacitors, we must have
The generalization to any number of capacitors is
3.2.2 Mixed connection
• Capacitors can also be connected in both series and parallel simultaneously.
• To solve mixed connected capacitors mathematically, first the capacitors
connected in one type must be solved and then the rest of the connection
can be sorted out easily.
Figure 3.6 Connecting capacitors in mix
(a) (b) (c)
3.3 Energy stored in a charged capacitor
• Many of the most important applications of capacitors depend on their
ability to store energy.
• The electric potential energy stored in a charged capacitor is equal to
the amount of work required to charge the capacitor —that is, to
separate opposite charges and place them on different conductors. Or,
• electric potential energy stored in a charged capacitor is also equal to
the total work done by the electric field on the charge when the
• When the capacitor is discharged, this stored energy is recovered as a work
done by electrical forces.
• The total work needed to increase the capacitor’s charge from zero to Q is
• Consider a parallel plate capacitor that is initially uncharged, so that the
initial potential difference is zero. After charging it, the final potential
difference across the capacitor reaches to ∆V. The average potential
difference during the charging process is
(work to charge a capacitor)
Then W is equal to the potential energy (U) of the charged capacitor. So we can
express U (which is equal to W) as below,
U is measured with joules
(c). Q1 + Q2 = Q0 Q1 = C1V and Q2 = C2V
• Most capacitors have a non-conducting material, or a dielectric, between
the two conducting plates.
• A common type of capacitor uses long strips of metal foil for the plates,
separated by strips of plastic sheet.
• A sandwich of these materials is rolled up, forming a unit that can provide
a capacitance of several microfarads in a compact package.
Figure 3.7 A common type of capacitor uses
dielectric sheets to separate the conductors.
• Placing a solid dielectric between the plates of a capacitor serves three
First, it solves the mechanical problem of maintaining two large metal
sheets at a very small separation without actual contact.
Second, using a dielectric increases the maximum possible potential
difference between the capacitor plates.
Third, the capacitance of a capacitor of given dimensions is greater when
there is a dielectric material between the plates than when there is vacuum.
• Suppose a capacitor has a capacitance when there is no material between
the plates. When a dielectric material is inserted to completely fill the
space between the plates, the capacitance increases to
is called the dielectric constant
We shall show that
is a m