UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
APSC 178: Electricity and Magnetism
MODULE 4.0
Capacitance and Dielectrics
Readings: Young and Freedman, Chapter 24
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
4.1 Capacitors and Capacitance
Any two conductors separated by an insulator forms a capacitor. In most applications, each conductor
initially has zero net charge and electrons are transferred from one conductor to the other; 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. We then say that the capacitor has a stored charge
Q.
The capacitor is represented by:
The electric field at any point between the conductors is proportional to the magnitude of charge on each
conductor. It follows that the potential difference is also proportional to Q. The ratio of the charge Q to the
potential difference V defines the capacitance:
The units of capacitance are in farads (F), equivalent to coulombs/volt (C/V).
Calculating Capacitance: Capacitors in Vacuum
The simplest form of a capacitor consists of two conducting plates, each with area A, separated by a distance
d. Let’s assume that we place a charge +Q onto the top plate, which will induce a negative charge –Q on the
bottom plate (to maintain overall neutrality). The charge on the top plate distributes itself to yield a charge
density of σ = Q/A. The electric field in the region between the plates is given by:
Since the field between the plates is uniform, the potential difference between the plates is:
From this, we can find the capacitance by taking a ratio of the charge Q to the potential difference V:
The constant ε0 is the permittivity of free-space which can now be expressed in units of F/m.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Example 1, Properties of a parallel plate capacitor: The plates of a parallel plate capacitor in vacuum are
5.00mm apart and 2.00 m2 in area. A potential difference of 10,000V is applied across the capacitor.
Compute the (a) capacitance; (b) the charge on each plate; and (c) the magnitude of the electric field between
the plates.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Example 2, A spherical capacitor: Two concentric spherical conducting shells are separated by vacuum. The
inner shell has a total charge of +Q and radius ra. The outer shell has charge –Q and radius rb. Find the
capacitance of the spherical capacitor.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Example 3**: The figure below represents the cross-sections of two spherical capacitors. Determine their
capacitances letting a = 1mm, b = 3mm, c = 2mm, r1 = 2.5, and r2 = 3.5.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
4.2 Capacitors in Series and Parallel
Capacitors in Series:
In a series connection, the magnitude of the charge on all the plates is the same. We can write the potential
differences between points a and c, c and b, and a and b as:
So the overall capacitance of the capacitors in series is given by:
The equivalent capacitance Ceq of the series capacitance is defined as the capacitance of a single capacitor
which could effectively replace the capacitors in series:
For the general case of several capacitors in series, we have that the reciprocal of the equivalent capacitance
of a series combination equals the sum of the reciprocals of the individual capacitances.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Capacitors in Parallel:
In a parallel connection, the potential difference for all individual capacitors is the same. Each capacitor will
induce charges Q1 and Q2. The total charge is a combination:
The parallel combination is equivalent to a single capacitor with the same total charge Q = Q1+Q2 and a
potential difference V. The equivalent capacitance is therefore given by:
It can be shown that the equivalent capacitance of a parallel combination equals the sum of the individual
capacitances.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Example 1, Capacitor Network: Find the equivalent capacitance of the capacitor network shown.
4.3 Energy Storage in Capacitors and Electric-Field Energy
We can calculate the potential energy U of a charged capacitor by calculating the work W required to charge
it. We know that in the final state, the capacitor will have a charge Q and a potential V. Here, we will let q
and v be the respective charge and potential in an intermediate step in the charging process.
The incremental work dW required to transfer dq is:
The total work W needed to increase the capacitor charge q from 0 to Q is found by integration:
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
If we define the potential energy of the uncharged capacitor to be zero, then W is equal to the potential
energy of the charged capacitor:
Electric-Field Energy
We can charge a capacitor by moving electrons directly from one plate to another. This requires doing work
against the electric field between the plates. Thus, we can think of energy as being stored in the field in the
region between the plates. Let’s find the energy per unit volume (energy density) between the plates:
Example 1, Q24.28: A capacitor of capacitance C is charged to a potential difference V0. The terminals of
the charged capacitor are then connected to those of an uncharged capacitor of capacitance C/2. Compute (a)
the original charge of the system, (b) the final potential difference across each capacitor, (c) the final energy
of the system, and (d) the decrease in energy when the capacitors are connected. (e) Where did the “lost”
energy go?
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Example 2: A spherical capacitor has charges of +Q and –Q on its inner (radius ra) and outer (radius rb)
conductors. Find the electric potential energy stored in the capacitor by using the capacitance and by
integrating the electric field energy.
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
4.4 Dielectrics
Most capacitors have a nonconducting material, or dielectric, between their conducting plates. Using a
dielectric increases the maximum possible potential difference between the conducting plates and increases
the capacitance of the capacitor relative to when the gap is filled with free space.
Let the original capacitance of a capacitor be defined as C0 = Q/V0 and let the modified capacitance in the
presence of a dielectric be C = Q/V. The ratio of C to C0 is called the dielectric constant of the material:
Induced Charge and Polarization
When a dielectric is inserted into a capacitor while the charge is kept constant, the potential difference
between the plates deceases by a factor K (or εr). Therefore, the electric field between the plates decreases
by the same factor:
Since the electric-field magnitude is smaller when the dielectric is present, the surface charge density must be
smaller as well. The surface charge on the conducting plates does not change, but an induced charge of
opposite sign appears on each surface of the dielectric. The dielectric was originally neutral and is still
neutral; the induced surface charges arise from a redistribution of positive and negative charge within the
dielectric – called polarization.
The product Kε0 is called the permittivity of the dielectric, denoted by ε:
In terms of ε we can express the electric field within the dielectric as:
The capacitance when the dielectric is present is given by:
The energy density u in an electromagnetic field when the dielectric is present is:
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UBCO School of Engineering
APSC 178
Prof. Kenneth Chau
Example 1, Q24.47: A 12.5 μF capacitor is connected to a power supply that keeps a constant potential
difference of 24.0V across the plates. A piece of material having a dielectric constant of K = 3.75 is placed
between the plates, completely filling the space between them. (a) How much energy is stored in the
capacitor before and after the dielectric is inserted? (b) By how much did the energy change during the
insertion? Did it increase or decrease?
Dielectric Breakdown
When any dielectric material is subjected to a sufficiently strong electric field, dielectric breakdown takes
place and the dielectric becomes a conductor. The maximum electric-field magnitude that a material can
withstand without occurrence of breakdown is called its dielectric strength.
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