Introduction. The are two forms of electricity, static electricity (electricity at rest) and dynamic electricity (electricity in motion). The branch of physics which studies static electricity is called electrostatics and the branch of physics which studies dynamic electricity is electrodynamics. Although most of the common uses of electricity involve electricity in motion, the fundamental concepts of electricity are best understood by first considering electricity at rest. Also since static electricity is by far the earliest known form of electricity, electrostatics is both historically and logically the best starting point in the study of electricity.
Early History of Electrostatics. The first record of static electricity was made by early Greek philosopher, Thales of Miletus (640-546 B.C.), about 600 B.C. He reports the fact that amber (called elektron in the Greek) would, when rubber with fur, attract light pieces of straw and other fibrous materials. He did not call this phenomenon electricity, because the word was not coined until many centuries later. In fact Thales did not know there was any such thing as electricity as we know it today. What he observed was called the amber phenomenon. The noun "electricity" was coined by Sir Thomas Browne in 1646 in his book Pseudoxia Epidemica (Vulgar Errors), in which he attacks many of pseudoscientific ideas of his day.
For a period of more than two thousand years after Thales, little was discovered concerning electricity. The first real systematic study of static electricity in modern times was not made until the latter part of the sixteenth century. Dr. William Gilbert (1544-1603), physician to Queen Elizabeth I and James I, published in London in 1600 a book on magnetism named De Magnete. In addition to an investigation of magnetism, he also records many facts and experiments on electricity. He found that many other materials besides amber, such as sulfur and glass, could also be electrified (amberized, as he called it) by rubbing. In the course of his experiments he observed that he was unable to electrified any of the metals. As a result of these experiments he divided all materials into two groups: electrics (from the Greek name for amber, elektron), those materials like amber which could be electrified by rubbing; and non-electrics those which could not. Later as a result of the experiments of Stephen Gray (d.1736) it was concluded that Gilbert's failure to electrify metals by rubbing them was a result of his failure to insulate them and thus any electricity caused by rubbing was immediately conducted away. Gray had discovered in 1729 that electricity could be transferred from one body to another. If an unelectrified body was connected by a metal wire to an electrified body, the electricity would travel along the wire to the unelectrified body. This phenomenon was called electrical conduction and such materials, usually metals, that would allow the electricity to flow and distribute itself all over the surface of a body made of them, no matter where the electricity was placed initially, were called conductors. The materials which Gilbert mistakenly called non-electrics are now known to be conductors. The other materials which Gilbert had called electrics that retained the electricity placed on the surface of a body made of them are now called insulators or non-conductors. Thus Gilbert's mistaken classification has been gradually superseded.
During the seventeenth and eighteenth centuries, many scholars discovered new electrical effects. Robert Boyle (1627-1691), in 1650, observed that in a vacuum electrified bodies would attract light materials. He concluded from this that the presence or absence of the electric effect did not depend on air. Boyle also discovered that other resins like had rubber, sealing wax, etc. could be electrified by triboelectrification, that is, by rubbing of one body against another. Later these materials were used to construct machines in which continuous rubbing would produce strong electrification accompanied by visible, audible sparks and by a violent stimulation of the human nervous system in contact with it. So amazing were these electrical effects that at one time public displays were widely attended, and private "electricity" parties were a popular form of entertainment.
In 1733, Charles Francois Dufay (1698-1739), a French physicist, observed that the resins, like sealing wax and amber, when rubbed with cat fur or wool, thus becoming electrified, differed from an electrified glass rod. The resins would strongly repel any electrified body which as attracted by electrified glass and attract any electrified body repelled by the glass. Thus Dufay concluded that there were apparently two kinds of frictional or static electricity. The electricity formed on rubbed glass and other vitreous substances he called vitreous electricity and the electricity formed on rubbed amber and other resins he called resinous electricity. Dufay observed as an immediate consequence of his discovery of the two kinds of electricity that two bodies with like kinds of electricity repel each other and two bodies with unlike kinds of electricity attract each other. This has become known as the law of charges or the first law of electrostatics.
In 1747, Benjamin Franklin (1706-1790), also recognizing two kinds of electricity, introduced the terms "positive" and "negative" to distinguish them. Thus, he said, we will arbitrarily call any electrified body positive if it is repelled by an electrified glass rod which has been rubber with silk, and we will call any electrified body negative if it is repelled by an electrified rubber rod which has been rubbed with cat fur. This law of signs is still used today to distinguish between positive and negative electricity.
Coulomb's Law of Electrostatics. About the middle
of the eighteenth century, measurements and mathematical procedures
were first applied to electrostatics. The result of this application
was a law which has been since called the second law of electrostatics
or Coulomb's Law. About 1760 the French mathematician
Daniel Bernoulli (1700-1782) suggested that the electrical force
of attraction and repulsion between two charged bodies varies
inversely as the square of distance between them. But his suggestion
was largely ignored. Later the English clergyman and schoolteacher
Joseph Priestley (1733-1804) proposed in 1767 that the force of
attraction between electrically charged bodies is subject to a
law similar to Newton's Law of Gravitation, but some 18 years
passed before it was experimentally verified. In 1795 a French
physicist, Charles Augustin Coulomb (1736-1806), using the torsion
balance, an instrument he had invented, measured the forces between
electrified bodies at rest that were small compared with distance
between their centers. Within the limits of accuracy of his measurements,
Coulomb showed that the force of attraction or repulsion between
two electrified bodies is directly proportional to the product
of their charges and inversely proportional to the square of distance
between their centers. This is known as Coulomb's Law of Electrostatics
and, expressed symbolically, is
F ∝ q1q2/r2,
or, expressed as a mathematical formula, is
F =
ke(q1q2/
r2), (1)
where F is the force of attraction or repulsion between
the two electrified bodies, whose charges or quantities of electricity
are q1 and q2, and r is the
distance between their centers. The constant of proportionality,
ke, depends for its value upon the units which are
used to measure force and distance, and the nature of the intervening
medium between the two electrified bodies. In the rationalized MKS
system of units, where force is measure in newtons and distance in meters,
the Coulomb's Law constant is
ke = 1/(4πε), (2)
where the Greek letter ε (epsilon) stands for absolute
permittivity and is by definition equal to the product of
ε0 and εr; that is,
ε = ε0εr, (3)
where ε0 is called the permittivity of empty space
or the permittivity of a vacuum, and
εr is called relative permittivity
or the dielectric coefficient or the dielectric constant
(sometimes represented by K).
The constant εr is a pure number (no units) and
depends only for its value upon the intervening medium between the two
charged bodies; it is greater than unity for all materials and is equal
to one in the case of a vacuum. The constant ε0 depends
upon the system of units employed in Coulomb's Law of Electrostatic.
In the rationalized MKS system of units ε0 has a value of
8.854 × 10-12 coulomb2 per
newton-meter2. Thus
F =
[1/(4πε)]
[(q1q2)/r2], (4)
where, in the rationalized MKS system of units, F is measured
in newtons, q1 and q2 are measured in
coulombs, and r is measured in meters. The proportionality constant
of Coulomb's Law, ke, has a value of
8.98776 × 109 (or 9 × 109)
newton-meter2 per coulomb2 in vacuum where
εr equals one. The unit for measuring electrical charge,
that is, the quantity of electricity, in the rationalized MKS system of units
is the coulomb. The coulomb is defined as that amount of charge on a
body which repels or attracts an equal amount of charge on another body place
one meter from it in vacuum, with a force of 9 × 109 newtons.
Coulomb's Law of Electrostatic as given in the above form is restricted in application to point charges, that is, to spherical shaped charged bodies whose dimensions are small compared with the distance between them. However, the law holds whether the point charges are like or unlike in polarity. If the appropriate signs are used for q1 and q2 in Coulomb's Law of Electrostatic, the sign of the force will be positive when the point charges have same sign thus indicating a force of repulsion. On the other hand, the sign of the force will be negative when the point charges are of opposite sign thus indicating a force of attraction.
Electrostatic Fields of Force. In describing the interaction of electrically charged bodies, Michael Faraday (1791-1867), in about 1840, noticed that an electrically charged body exerts a force on another charged body anywhere in the space surrounding it. To indicate the regional action of the electrostatic forces, he introduced the concept of field of force or, more briefly, the field. A field of force is all the space in which a force is detected. The region theoretically extended to infinity, but practically it has finite dimensions. Thus using this concept of a field of force, an electric field of force, or simply electric field, may be defined as that area or region in which an electrically charged body will have a force of attraction or repulsion exerted on it. Note that nothing is said about the source of this field. Our criterion of the existence of an electric field at a point in space is whether a charged body place at a point in that space will experience a force acting on it. If it does, the electric field by definition exists at that point no matter what is causing it. If the source of an electric field of force is stationary or is a static electrically charged body, then it is called an electrostatic field. Thus an electrostatic field of force may be defined as that region or area about a stationary electrically charged body in which another electrically charged body will have force of attraction or repulsion exerted on it. As a field of force, an electrostatic field has both direction and magnitude.
The direction of an electric field of force at any point is defined as the direction of the force exerted on a positive test charge placed at that point. A test charge is a positive point charge used for testing the direction and magnitude of an electric field of force. As such the test charge is an imaginary, mathematical device introduced in order to describe an electric field of force. The direction of an electric field of force at any point then is said to be the same as the direction of the force that would be exerted on a test charge if placed at that point. Even though such a test charge is imaginary and hypothetical, it is a useful and convenient mathematical device for describing an electric field. Notice that a positive point charge is the criterion. This is by arbitrary and universal agreement.
The magnitude of an electric field of force at any point is defined
as the ratio of the force exerted on a positive test charge placed
at that point to the quantity of charge of the test charge. If
Ft stands for the force exerted on the test charge and
qt stands for the quantity of electric charge on the
test charge, then the defining formula for the magnitude of an electric
field of force is
E = Ft/qt, (5)
where E stands for the magnitude of the field and is called
electric field intensity or electric field strength.
In the rationalized MKS system of units where force is measure
in newtons and the quantity of electric charge in coulombs, electric
field intensity is measured in newtons per coulomb. Thus, the
magnitude of an electric field is the force per unit charge.
Since the force exerted on the positive test charge is a vector
quantity, electric field intensity is also a vector quantity.
The direction of this quantity is the same as the direction of
the field of force and the magnitude of this quantity is the same
as the magnitude of the field. In an electrostatic field, the
force exerted on the positive test charge may be calculated by
Coulomb's Law. Hence, the electric field strength E about
an isolated point charge can be calculated.
E = Ft/qt =
Ft(1/qt) =
[1/(4πε)][(qtQ)/r2]
(1/qt) =
[1/(4πε)][Q/r2], (6)
where Q is the quantity of electric charge of an isolated
point charge which is the source of the electric field and r
is the distance from the isolated source point charge Q
to the point where the electric field intensity is to be calculated.
Notice that the magnitude of the charge of the point test charge
does not enter into this formula (6), emphasizing what was said
above about the test charge being an hypothetical mathematical
device. This formula implies that the electric field intensity
at points near the source charge will be greater than at points
more distant. The electrostatic field grows weaker the further
a point in the electrostatic field is from the source of the field.
In general this statement is true no matter what the source of
the field. However, the above formula (6) holds true exactly
only for fields about an isolated point charges. The methods
of the calculus must be used to find the electric field intensity
of fields about sources other than isolated point charges.
In the attempt to represent this electric field of force, Faraday introduced another concept, lines of force. By drawing lines in the region of the field in a certain way, a graphical representation of the field may be obtained. These lines of force represent two things concerning the field - its direction and its magnitude.
Lines of force represent the direction of an electric field by being drawn in such a way that the direction of the lines at a point is the direction of the field at that point. The direction of a line of force at a point is the direction of the tangent to it at that point. Since at any point in an electric field, the field can have but one direction, only one line of force can be drawn through each such point of the field. In other words, lines of force can never intersect, for this would imply that the field would have two directions at the point of intersection.
The lines of force also represent the magnitude of an electrostatic
field. This is accomplished by spacing of the lines in such a
way that the number of lines per unit area crossing a surface
perpendicular to the direction of the field is equal to the product
of absolute permittivity ε and electric field intensity E
at that surface. If the Greek letter psi (ψ) is used to
designate the number of lines of force and if A is the symbol
for the area of a surface perpendicular to the direction of the field,
then the above rule may be expressed by the following formula.
ψ/A = εE. (7)
By use of this rule, the number of lines of force drawn to represent
the electric field are limited in such a way as to represent the
magnitude of the field. By making the number of lines of force
per unit area of a surface perpendicular to the direction of the
lines proportional to the electric field intensity, a suitable
limiting of the number of lines to be drawn to represent the field.
Thus the lines of force may be used to represent the magnitude
of the field as well as its direction. Practically, this rule
means that in a region where the field intensity is large, the
lines of force will be closely spaced, and the region where the
field intensity is small, the lines will be widely separated.
A new physical quantity was introduced in the
formula (7)
given above. It is called electric flux, and was symbolized
by the Greek letter ψ. In the rationalized MKS system of units,
electric flux is measured in coulombs. A single line of force
or flux is called a coulomb. Now it will be noticed that the
coulomb is also the unit by which electric charge is measured.
This is no accident. An example will show the reason for using
the same unit for measuring these two physical quantities. Consider
the electric field about an isolated point charge with a charge
of 14 coulombs. The electric field intensity at all points one
meter from this isolated point charge would be same numerically
and can be calculated by the
formula (6)
above.
E = [1/(4πε)](Q/r2) =
[1/(4πε)](14/12) =
14/(4πε) newtons per coulomb.
All of these points that are one meter from the isolated point
charge constitute a sphere whose center would be at the center
of the isolated point charge. This spherical surface would everywhere
be perpendicular to the direction of the field. The total area
of this spherical surface may be calculated by the formula
A = 4πr2 = 4π(1)2 = 4π
square meters,
since the radius of the sphere is one meter. Now by solving the
formula (7)
for electric flux and substituting in the values
just obtained for the electric field intensity and the area of
spherical surface, the number of lines that could be drawn through
the spherical surface may be found.
ψ = εEA = ε[14/(4πε)](4π) =
14 lines of force.
It is to be noted that the number of lines of force that could
be drawn through a spherical surface whose center is at the center
of a isolated point charge would be the same no matter what is
the radius of the spherical surface. The number of lines of force
shown radiating out from or converging on an isolated point charge
is independent of the radius of the concentric spherical surface.
That is,
ψ = εEA =
ε[1/(4πε)](Q / r2)
(4πr2) = Q. (8)
Thus the total number of lines of force ψ radiating out from or
converging on an isolated point charge through a spherical surface
enclosing the charge depends only upon and is equal to the quantity
of electric charge Q within the enclosing surface, that is,
ψ = Q.
And since the quantity of electric charge Q is
measured in coulombs and electric flux ψ is numerically equal to it,
electric flux is also measure in coulombs. This is a form
of the law that was discovered by the Karl Friedrich Gauss (1777-1855),
called Gauss' Law for electric fields. The spherical surface
enclosing the charge is called a Gaussian surface.
Since the concept of electric flux has been introduced, another
physical quantity may now be defined. This physical quantity
is called electric flux density and it is defined as the
number of lines of electric flux per unit area threading through
a surface perpendicular to the direction of the field. Electric
flux density is symbolized by the letter D. Thus the defining
formula for electric flux density is
D = ψ/A. (9)
In the rationalized MKS system of units, electric flux density
is measure in coulombs per square meter, since the area is measured
in square meters nd the charge in coulombs . Electric flux density
is a vector quantity whose direction is the same as the direction
of the electric flux (the electric lines of force), that is, the
direction of the electric field.
Using this definition of electric flux density and
formula (7),
the relation between electric flux density and electric field intensity
may be derived.
D = ψ/A = εE. (10)
The direction of the vector quantity of electric flux density
will be same as the direction the vector quantity of electric
field intensity.
In conclusion, an electric line of force may now be defined as an imaginary line drawn in such a way in a diagram of an electric field of force so that its direction at any point represents the direction of the field at that point and the spacing between it and adjacent lines represents the magnitude of the field.
The electrostatic fields about an isolated point charges or combinations of several point charges are called non-uniform electrostatic fields. In such fields the closer to the source of the field, the stronger the electric field intensity. However, in the electrostatic field between two parallel, oppositely charged plates, the electric field intensity has the same value for all points between the two plates except for points at the edges. The lines of force between these two parallel, oppositely charged plates are straight, parallel lines uniformly space except at the edges. If the plates are sufficiently close, then this spreading or "fringing" of the field at the edges of the plates may be neglected. Such a field is called a uniform electrostatic field.
The electric field intensity at any point between the two parallel,
oppositely charged plates except at the edges may be found by
the following formula.
E = (1/ε)(Q/A), (11)
where Q is the charge on either plate measured in
coulombs and A is the area of either plate measured
in square meters. Notice that the factor 4π does not occur in
this formula because it was introduced into the constant of proportionality
in Coulomb's Law of Electrostatics which, while fundamental, is
less frequently used, and it is eliminated in the derivation of
this formula (11), which is more frequently used. Systems of
units that relegate the factor 4π to formulas which, while more
fundamental, are less frequently used so that it will not occur
in formulas more frequently used, are called rationalized
system of units.
Electric Energy and Electric Potential Energy. In the preceding section electrostatic fields have been described in terms of the forces exerted on a positive test charge. The convention of lines of force has been used to graphically represent the field so described. There is another way in which an electrostatic field may be described which in some instances is a much more useful tool for handling a large group of engineering problems. An electrostatic field may be describe in terms of the energy or the ability to do work by the electric forces of the field.
In mechanics energy was defined as the ability to do work. And in mechanics work is done if a force moves a body through a distance and the amount of work done can be found by multiplying the force times the distance through which the force moved the body. In the MKS system of units the unit of work is the joule and it is defined as the work done when one newton of force acts through a distance of one meter. Because of the close connection between energy and work, energy is measure in the same units as work, which in the MKS system of units is the joule. Now the energy of a body is the ability of that body to do work, that is, the ability of a body to exert a force on another body that would move that body through a distance. If a body possessed this energy by virtue of its motion, this energy was called kinetic energy. However, if a body possesses energy as a result of its position in a gravitational field, then the body is said to possess gravitational potential energy. The gravitational potential energy that a body possesses because of its position may be found by calculating the amount of work necessary to get it to that position from another position called the reference position or level or by calculating the amount of work done by the gravitational force in returning the body to the reference level. With a little thought it will be seen that, on the one hand, kinetic energy will result in work because of contact forces while, on the other, gravitational potential energy is a result of work being done against or by forces acting-at-a-distance (field forces).
Electrical energy may also be defined in much the same way as mechanical energy. Electrical energy is the ability of an electric charge to do work, that is, the ability to exert a force that would move another electric charge through a distance. Two kinds of electrical energy may also be distinguished; kinetic electric energy and potential electric energy. Kinetic electric energy is that energy possessed by an electric charge because of its motion. The motion of the electric charge could result in work if it should exert a force on another electric charge by virtue of its motion. Potential electric energy is the energy possessed by an electric charge as a result of the position it has in an electric field of force.
The amount of electric potential energy that an electric charge
would have because of its position in the electric field may be
found by calculating the amount of work necessary to move it to
that position from another position which is taken as reference
position or by calculating the amount of work done by the electric
force in returning the body to the reference position. The reference
position in an electrostatic field of force is taken, for its
mathematical simplicity, to be at a point an infinite distance
away from the source of the field. Consider, for example, the
electrostatic field about an isolated positive point charge, Q,
which is the source of the electrostatic field.
A test charge is placed at point b. (A test charge is
a positive point charge whose electrical charge is sufficiently
small so that its presence in the field will not alter it.)
The absolute electric potential energy of the test charge
at the point b in the electrostatic field may be defined
as the work done against the force exerted on it by the field
when the test charge is brought from an infinite distance away
(at infinity) to the point b (presuming the source of the
field Q to be a positive numerical value). That is,
P.E.b = Ub =
-W∞→b, (12)
where P.E.b and Ub are the symbols for
electric potential energy.
Since the force on the test charge is not constant Ft,
the electrostatic force would gradually increase as the electric test charge
qt would move closer to the source of the field Q,
the work done against this force cannot be found by simply multiplying the
force by the displacement. This can be done only if the force is constant.
The work done against the forces of the electrostatic field as the test charge
moves from infinity to the point b can be found by the calculus.
Accordingly, it is found that
P.E.b = Ub =
-W∞→b =
[1/(4πε)](Qqt /
rb), (13)
where Q is the magnitude of the isolated point charge
that is the source of the field, measured in coulombs, and
qt is the magnitude of the test charge measured in
coulombs, and rb is the distance that the point
b is from the source of the field measured in meters. Absolute
electric potential energy is measure in joules in the rationalized MKS
system of units. According to formula (13), the work done in bringing
the test charge from infinity to the point b in the electrostatic
field is entirely independent of the path taken by the test charge.
It depends only on its final position in the electrostatic field.
In defining absolute electric potential energy, the reference
position was for mathematical simplicity taken to be a point infinite
distance away from the source of the field. However, because
of the difficulty of giving a physical meaning to absolute electric
potential energy in practical engineering problems, the reference
position may then be taken at any convenient point in the field.
The electric potential energy would then be relative to that
arbitrarily chosen reference position. Consider again the electrostatic
field about an isolated positive point charge Q.
P.E.ba = Uba =
-Wa→b. (14)
This kind of electric potential energy may be called relative
electric potential energy. In the case of the electrostatic
field about an isolated positive point charge, Q, the relative
electric potential energy of a test charge qt at point
b relative to point a, would be given by the following
formula which was found by means of the calculus.
P.E.ba = Uba =
-Wa→b =
[Qqt / (4πε)][(1 / rb) -
(1 / ra)], (15)
where ra is the distance measured in meters that the point
a is from the source of the field, and rb is the
distance measured in meters that the point b is from the
source of the field. If the brackets are removed, the formula (15)
will take the following form.
P.E.ba = Uba =
-Wa→b =
[Qqt/(4πεrb)] -
[Qqt/(4πεra)], (16)
where [Qqt/(4πεrb)]
is the absolute electric potential energy possessed
by the test charge when at point b and
[Qqt/(4πεra)]
is the absolute electric potential energy possessed by the test
charge when at point a. Thus the relative electric potential
energy of a test charge when at b relative to the reference position
at point a is equal to the difference between the
absolute electric potential energy at point b and
the absolute electric potential energy at the reference
position, point a. Letting Ub be the absolute
electric potential energy at point b and Ua be the
absolute electric potential energy at point a, then the relative
electric potential energy at point b relative to point a
P.E.ba or Uba is
P.E.ba = P.E.b - P.E.a
or Uba =
Ub - Ua. (17)
Thus relative electric potential energy may be called electric
potential energy difference because it is the difference between
the absolute potential energies of those two points.
Electric Potential. In order to indicate how such
electric potential energy is available at a point in an electric
field per each unit of charge, the concept of electric potential
has been introduced. Electric potential may be defined in much
the same way in which gravitational potential was defined earlier.
The electric potential at a point in a electric field is the
ratio of the electric potential energy of a test charge at that
point to the magnitude of the test charge. That is, the electric
potential at a point in an electric field is the electric potential
energy per unit charge of a test charge placed at that point.
The defining formula for electric potential is given by the following
equation.
V = P.E./qt = U/qt =
-Wt/qt, (18)
where Wt and U is the electric potential energy
of the test charge at a point in an electric field, qt
is the magnitude of the charge of the test charge, and V
is the electric potential at the point. When a subscript is upon
the symbol V as, for example, Va or
Vb, it indicates the electric potential is the
electric potential of a specific point. In the rationalized MKS system
of units, where electric potential energy is measured in joules and electric
charge is measured in coulombs, electric potential according to
the defining
formula (18)
above is measured in joules per coulomb.
One joule per coulomb is called one volt. The volt is
named in honor of the Italian scientist Alessandro Volta, who
invented the "voltaic pile," the first electrolytic
cell. The electric potential at a point in an electric field
is one volt, when the ratio of the electric potential energy of
a test charge at the point to the magnitude of the test charge
is one joule per coulomb. Accordingly, this electric potential at a point
in an electric field is often called the voltage at the point.
Some useful submutiples of the volt are the millivolt (mv)
which is one-thousandth of a volt and microvolt (μv)
which is one-millionth of a volt. Some useful multiples of the volt
are the kilovolt (kv) which is one thousand volts and the
megavolt (Mv) which is one million volts. Since
electric potential energy is a scalar quantity, electric potential
is also a scalar quantity; it has magnitude but no direction.
Note that electric potential like gravitational potential is the property of a point in the electric field and not of the electric charge placed at the point. It is a way of indicating how much electric potential energy U is available at that point for each unit of electric charge, if an electric charge were placed there. That is, point in the electric field would have an electric potential whether there was an electric charge there or not. The test charge used in the definition of electric potential given above is an imaginary, hypothetical electric charge which has been introduced just for the purpose of defining electric potential.
Since there are two kinds of electric potential energy, it follows
from the definition above that there are two kinds of electric
potential - absolute electric potential and relative electric
potential. The absolute electric potential at a point
in the electrostatic field is the ratio of absolute potential
energy of a test charge at that point to the magnitude of the
test charge. Consider again the electrostatic field about an
isolated positive point charge, Q. A test charge placed
at point b will possess a certain amount of absolute electric
potential energy which may be calculated by
formula (13).
According to the definition just given, the absolute electric potential
at the point b would be equal to the ratio of the absolute
electric potential energy of a test charge at point b to
the magnitude of the test charge. That is,
Vb =
P.E.b/qt =
Ub/qt. (19)
By substitution and algebraic simplification, the formula for
the absolute electric potential at point b in the electrostatic
field is obtained.
Vb = Ub(1 / qt) =
[Qqt/(4πεrb)]
(1/qt) =
Q / (4πεrb). (20)
This formula can be used to find the absolute electric potential
at any point in the electrostatic field of any isolated point charge.
The relative electric potential of a point in an electric
field is the absolute electric potential of that point relative
to some other point which is taken as a reference position. It
is equal to the ratio of the relative electric potential energy
of a test charge at that point to the magnitude of the test charge.
Consider still again the electrostatic field about an isolated
positive point charge, Q. A test charge placed at point
b will possess a certain amount of relative electric potential
energy which is relative to some reference position, point a.
This relative electric potential energy may be calculated by
formula (15).
According to the definition just given, the relative
electric potential of the point b to the reference point
a is equal to the ratio of the relative electric potential
energy of a test charge at point b to the magnitude of
the test charge.
Vba =
P.E./qt =
Uba/qt. (21)
By substitution and algebraic simplification, the formula for
the relative electric potential at point b to the reference
point a in the electrostatic field is obtained.
Vba =
[Qqt / (4πε)][(1 / rb) -
(1 / ra)](1 / qt) =
[Q/(4πε)][(1/rb) -
(1/ra)]. (22)
This formula can be used to find the relative electric potential
of any point in an electrostatic field of any isolated point charge
relative to any arbitrarily chose reference position.
If the brackets are removed, formula (22) will take the following form.
Vba =
Q/(4πεrb) -
Q/(4πεra), (23)
where Q/(4πεrb)
is the absolute electric potential of point b and
Q/(4πεra)
is the absolute electric potential of point a. The relative
electric potential of point b to the reference point a
is equal to the difference between the absolute electric potential
of point b and of the reference point a.
Vba =
Vb - Va. (24)
Thus relative electric potential is called electric potential
difference.
Electric potential difference between two points in an electrostatic
field is basically equal to the work done per unit charge against
the forces of the electric field as the charge is moved from one
point to the other. A point in an electrostatic field is said
to be at a higher absolute electric potential than another point
in the field only if work is done against the forces of the field
as a positive charge is moved from the other point to that point.
That is, the point is at a higher potential than the other point
if a positive charge gains electric potential energy as it so
moved from the other point to that point. A positive charge will
gain electric potential energy only if it is moved against the
direction of the field. Otherwise, it will lose electric potential
energy if it moves in the direction of the field. A point in
an electrostatic field is said to be at a lower absolute electric
potential than another point in the field only if a positive charge
loses electric potential energy as it moves from the other point
to that point. The electric potential difference between two
points in an electrostatic field will be positive if the
first point is at a higher absolute electric potential than the
second point (first and second points refers to
the order of subscripts on the symbol for potential).
Vba > 0,
if Vba = Vb - Va and
Vb > Va. (25)
A positive charge will gain electric potential energy as it moves
from second point a to the first point b. The electric
potential difference between two points in an electrostatic field
will be negative if the first point is at a lower absolute
electric potential than the second point.
Vba < 0,
if Vba = Vb - Va and
Vb < Va. (26)
A positive charge will lose electric potential energy as it moves
from second point a to first point b.
If a positive charge gains electric potential energy as it moves
from a point of lower absolute electric potential to a point of
higher absolute electric potential, then it will lose that electric
potential energy as it moves back from the point of higher absolute
electric potential to the point of lower absolute electric potential.
As the positive charge moves from the point of lower absolute
electric to a point of higher electric potential not only does
it gain electric potential energy but it is moving through a positive
electric potential difference.
Vba > 0,
if Vba = Vb - Va and
Vb > Va. (25)
As the positive charge moves from the point of higher absolute
electric potential to the point of lower absolute electric potential
not only does it lose electric potential energy but it is moving
through a negative electric potential difference.
Vab > 0,
if Vab = Va - Vb and
Va > Vb. (27)
It follows from the gain and loss of electric potential energy
of the positive charge as it moves from the point of lower absolute
electric potential a to the point of higher absolute electric
potential b and back again that
Vba + Vab = 0, (28)
that is, that the sum of the electric potential energy gained
per unit charge, Vba , and the electric potential energy
lost per unit charge, Vab, is equal to zero. Absolute
electric potential at a point in an electrostatic field may be
looked upon as the electric potential difference between the point
and a point at infinity where the absolute electric potential
is arbitrarily assumed to be zero. Thus in
formula (23),
if ra is infinite, the second term of the right side
of the formula becomes zero. Thus electric potential difference
Vba becomes the absolute electric potential at point
b, Vb.
Absolute electric potential and electric potential difference are both measured in volts in the rationalized MKS system of units. The absolute electric potential at a point in an electrostatic field is one volt if one joule of work is done against the forces of the field for each coulomb of electric charge brought from infinity to that point. The electric potential difference between two points in an electrostatic field is one volt if one joule of work is done against the forces of the field for each coulomb of electric charge brought from the second point to the first point.
Consider a charged metal sphere whose radius is r. It
can be shown that the absolute electric potential at all points
within this charged metal sphere is constant and equal to the
absolute electric potential at its surface. The absolute electric
potential at its source can be calculated by form (20), if the
distance rb is taken to be equal to the radius
r of the sphere.
Vs = Q/(4πεr), (29)
where Vs is the absolute electric potential at the surface
of the sphere whose radius is r, and Q is the magnitude
of the electric charge that is evenly distributed over the entire
surface of the metal sphere. The absolute electric potential
at any point in the electrostatic field outside of the charged
metal sphere may be found by
formula (20)
where rb is never less than the radius r
of the sphere.
Consider two oppositely charged parallel plates a distance d
apart measured in meters. As it was pointed out earlier the
electrostatic field between two oppositely charged parallel plates
is uniform, that is, the electric field intensity everywhere is
of the same value, except near the edges, if the plates are large
and fairly close together. The electric field intensity in this
uniform field may be calculated by
formula (11).
A test charge qt will therefore have a force exerted on it
by the electrostatic field which will everywhere in the field be constant
in direction and magnitude. According to the defining formula of electric
field intensity,
formula (5),
when it is solved for Ft,
the force exerted on the test charge will be equal to the product
of the magnitude of the test charge qt and the electric
field intensity E.
Ft = qtE. (5b)
The work necessary to move the test charge from the negatively
charged plate b to the positively charged plate a against
the field will be equal to the product of the magnitude of the
test charge qt, the electric field intensity E,
and the space s between the plates. That is, since
W = Fs = -Fts then
-Wb→a =
-(-Fts) =
qtEs. (30)
This work is equal to the gain in electric potential energy of
the test charge and the electric potential energy difference between
the positively charged plate a and the negatively charged
plate b.
P.E.ab = Uab =
-Wb→a =
qtEs. (31)
According to the definition of electric potential difference,
and by substitution and algebraic simplification, the formula
for the electric potential difference between the two plates is
obtained.
Vab =
P.E.ab/qt =
qtEs/qt =
Es. (32)
Since the electric field intensity anywhere between these two
oppositely charged parallel plates may be found by
formula (11),
then by substitution, formula (32) will take the following form.
Vab = Qs/εA, (33)
where Q is the electric charge on either plate, regardless
of sign, measured in coulombs, s is the space between
the two parallel plates measure in meters, A is the area
of the surface of either plate measured in square meters, and
Vab is the electric potential difference between two
plates measured in volts. This electric potential difference between
the two plates is also called the voltage difference. Solving
formula (33) for Q, we find that the charge on either plate is
directly proportional to the electric potential difference
Vab between the two plates. That is,
Q = (εA/s)Vab =
CVab, (34)
where the constant of proportionality C = εA/s
is called capacitance and is measure in farads.
Equipotential Lines and Surfaces. A graphical picture of the condition of the absolute electric potentials which may exist in the electric field about a charged body may be obtained by the use of equipotential lines and surfaces. Equipotential lines are a graphical representation in two dimensions of the absolute electric potential distribution of an electric field, and equipotential surfaces are a graphical representation in three dimensions of the absolute electric potential distribution of an electric field. Equipotential lines and surfaces represent the absolute electric potential distribution of an electric field as electric lines of force represent the electric field intensity variations of an electric field. An equipotential line is an imaginary line drawn in such a way in a plane of an electric field such that every point on it has the same value of absolute electric potential. An equipotential surface is an imaginary surface constructed in such a way in the space of an electric field that every point on it has the same value of absolute electric potential. It may be possible to construct equipotential lines and surfaces through every point of an electric field; however, it is customary to show only a few of the equipotentials in a diagram. Those equipotentials that are shown in a diagram are chosen so that they are separated by equal intervals of absolute electric potential but not necessarily equal intervals of distance. These equal intervals are chosen arbitrarily and for convenience.
Consider the non-uniform electrostatic field about an isolated positive point charge, q. According to formula (20) all points the same distance from the source of the field Q will have the same value of absolute electric potential. All these points will constitute an imaginary sphere at whose center is the source of the field. Because the absolute electric potentials at all points on its surface are the same value, this imaginary sphere is called an equipotential surface. This spherical equipotential surface is everywhere perpendicular to the direction of the field. Hence, the lines of force which represent the direction of the field are everywhere perpendicular to the equipotential surfaces which represent the absolute electric potential distribution of the field. In general, the lines of force of a field are curved or straight lines and the equipotential surfaces are curved surfaces. The lines of force and the equipotential surfaces in a field will form a set of mutually perpendicular lines and surfaces. In the special case of a uniform field, the lines of force will be parallel straight lines and the equipotential surfaces will be parallel planes perpendicular to the lines of force. To move a test charge over an equipotential surface no work would be done and the test charge would not gain or lose electric potential energy. Work would be done and electric potential energy would be gained or lost only if the test charge moves against or with the field or at an angle to the field less than ninety degrees.