2.1.ELECTRIC CHARGE
Charge is a fundamental and characteristic ptoperty of the microscopic particles that make up matter. In fact, all atoms are composed of protons, neutrons, and electrons, and two of these particles bear charges. But even though on a microscopic scale matter is composed of a large number of charged particles, the powerful electrical forces associated with these particles are fairly well hidden in a macroscopic observation. The reason is that there are two kinds of charge, positive and negative, and an ordinary piece of matter contains approximately equal amounts of each kind. From the macroscopic viewpoint, then, charge refers to net charge, or excess charge. When we say that an object is charged, we mean that it has an excess charge, either an axcess of electrons (negative) or an acsess of protons(positive). It charge will usually be denoted by the symbol q.
2.2.COULOMB’S LAW
In the mks system, Coulomb’s law for the force between two point charges can thus be written
F1 =
r12 r1
r2
0
We may describe a charge distribution in terms of a charge density function, defined as the limit of the charge per unit volume as the volume becomes infinitesimal. Care must be used, however, in applying this kind of description to atomic problems, since in such cases only a small number of electrons are involved, and the process of taking the limit is meaningless. Leaving aside atomic cases. We may proceed as though a segment of charge might be subdivided indefinetely. We thus describe the charge distribution by means of point functions.
A volume charge density is defined by
and the surface charge density is defined by
If charge distributed through a volume V with a density , and on the surface S that bounds V with a density , then the force exerted by this charge distribution on a point charge q located at r.
Fq =
2.3.THE ELECTRIC FIELD
The electric field at a point is defined operationally as the limit of the force on a test charge placed at the point to the charge of the test charge-the limit being taken as the magnitude of the test charge goes to zero. The customary symbol for the electric field is E.
In vector notation, the definition of E becomes
E =
2.4.THE ELECTROSTATIC POTENTIAL
A direc calculation shows that
The electrostatic potensial due to a point charge q, is
Which is readily verified by direct differentiation. Thus, the potensial that corresponds to the electric field.
Which is also easily verified by direct differentiation
2.5.CONDUCTORS AND ISOLATORS
So far as their electrostatics behavior is concerned, materials may be divided into two categories, conductors of electricity and insulators(dielectrics). Conductors are subtance, like the metals, which containlarge numbers of essentially free charge carriers. These charge carriers (electrons in most cases) are free to wander throughout the conducting material,they respond to almost infinitesimal electric field, and they continue to move as long as they experience a field. These free carriers carry the electric current when a steady electric field is maintained in the conductor by an external scurce of energy.
Dielectrics are subtance in which all charged particles are bound rather strongly to constituents molecules. Strictly speaking, this definition applies to an ideal dielectric-that is, on showing no conductivity in the presence of an externally maintained electric field. Real physical dielectrics show a feeble conductivity, but in a typical dielectric, the conductivity is 1020 times smaller than that of a good conductor. Since 1020 is a tremendous factor, it is usually sufficient to say that dielectrics are nonconductors.
2.6.GAUSS’S LAW
An important relationship exists between the integral of the normal component of the electric field over a closed surface and the total charge enclosed by the surface. This relationship, known as Gauss’s law, will now be investigated in more detail. The electric field at point r due to a point charge q located at the origin is
E(r) =
Consider the surface integral of the normal component of this electric field over a closed surface that encloses the origin and, consequenly, the charge q
If S is a closed surface that bounds the volume V
Which is known as Gauss’s law
2.7.APPLICATION OF GAUSS’S LAW
In order the Gauss’s law be useful in calculating the electric field, it must be possible to choose a closed surface such that the electric field has a normal component that is either zero or a single fixed value at every point on the surface. As an example, consider a very long line charge of charge density λ per unit lenght. The cylindrical surface contributes 2 r since E is radial and independent of the position of the cylindrical surface. Gauss’s law the takes the form
2 r =
Er =
From this result, the potential can be obtained by integration:
= ln r + C
2.8.THE ELECTRIC DIPOLE
Two equal and opposite charges separated by a small distance form an electric dipole. The electric field and potential distribution produced by such a charge configuration can be investigated with the aid of the formulas of section 2.3 and 2.4. the electric field at r is found to be
E(r) =
2.9.MULTIPOLE EXPANSION OF ELECTRIC FIELDS
It is apparent from the definition of dipole moments given previously that certain aspects of the potential distribution produced by a specified distribution of charge might well be expressed in terms of it’s electric dipole moment. In order to do so, it is necessary to define the electric dipole moment of an orbitary charge distribution. Rather than make motivated definition, we shall consider a certain expansion of the electrostatic potensial due to an arbitary charge distribution. To reduce the number of position coordinates, a charge distribution in the neighborhood of the origin the charge distribution can be entirely enclosed by s sphere of radius a, which a small a compared with the distance to the point of obsevation
2.10.THE DIRAC DELTA FUNCTION
The dirac delta function δ(r) can serve this purpose, and in addition it is valuable mathematical tool in many calculations. We write
(point charge)
where
δ(r) = 0 for r
and
Some other properties of the delta function can be obtained as consequences of Gauss’s law in differential form:
for a point charge q at r = 0
or
Also, since
then
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