4.1.POLARIZATION
Since depends on the size of the volume element, it is more convenient to work with P, the electric dipole moment per unit volume. P is usually called the electric polarization, or simply the polarization, of the medium.
P =
Strictly speaking P must be defined as the limit of this quantity as v becomes very small from the macroscopic viewpoint. In this way P becomes a point function,P(x,y,z)
4.2.FIELD OUTSIDE OF A DIELECTRIC MEDIUM
Where the volume integral of has been replaced by a surface integral through application of the differgence theorem, and n is the outward normal to the surface element da’ (outward means out of the dielectric).
The quantities P.n and - that appear in the integrals of Eq. Above are two scalar function obtained form the polarization P.It is expedient to give these quantities special symbols. Since they have the dimension charge per unit area and charge per unit volume, respectively, we write
p = P.n = Pn
and
p = -
we call p and p polarization charge densities
4.3.THE DIELECTRIC FIELD INSIDE A DIELECTRIC
The(macroscopic) electrik field is the force per unit charge on a test charge embedded in the dielectric, in the limit where the test charge is to small that it does not itself affect the charge distribution.
The electrostatic field in a dielectric must have the same basic properties that we found applied to E in vacuum, in particular, E is a conservative field and, hence, derivable from a scalar potential. Thus
x E = 0
or, equivalenly,
The electric field in a dielectric is equal to the electric field inside a needle-shaped cavity in the dielectric, with the cavity axis oriented parallel to the direction of the electric field.
4.4.GAUSS’S LAW IN A DIELECTRIC: THE ELECTRIC DISPLACEMENT
where Q is the net emboded charge
Q = q1 + q2 + q3
and Qp is the polarization charge
Qp =
Qp =
We define, therefore, a new macroscopic field vector D, the electric displacement:
D =
Which evidently has the same unit as P, charge per unit area
This result is usually referred to as Gauss’s law for the electrical displacement, or simply Gauss’s law
4.5.ELECTRIC SUSCEPTIBILITY AND DIELECTRIC CONSTANT
These results are summarized by the constitutive equation
P = x(E)E
Where the scalar quantity x(E) is called the electric susceptibility of the material.
D = (E)E
(E) = 0 + x(E)
Where (E) is the permitivity of the material. It is evident that , 0 and x all have the same units.
In other words, x and are frequently constants characteristic of the material. Materials of the type will be called linear dielectrics, and they obey the relations
P = xE
D = E
The electrical behavior of a material is now completely specified by either the permitivity or the susceptibility x. It is more convenient, however, to work with a dimensionless quantity K defined by
= K 0
K is called the dielectric coefficient, or simply the dielectric constant
K =
4.6.POINT CHARGE IN A DIELECTRIC FLUID
q = r2D
the electric field and polarization may now be evaluated quite easily:
E =
P =
Thus, the electric field is smaller by the factor K than would be the case if the medium were absent.
Our point charge q is a point in the macroscopic sense. Suppose it is large on a molecular scale and we can assign to it a radius b, which eventually will be made to approach zero. The total surface polarization charge is then given by
Qp =
The total charge
Qp + q =
4.7.BOUNDARY CONDITIONS ON THE FIELD VECTORS
Let us construct the small pilbox-shaped surface S that intersect the interface and encloses an area S of the interface, the height of the pillbox being negligibly small in comparison with the diameter of the bases. The charge enclosed by S is
But the volume of the pillbox is negligibly small, so that the last term may be neglected. Applying Gauss’s law to S, we find
or
(D1 – D2) . n2 =
If there is no external charge in the region, then Q = 0, and the same amuont of flux enters the tube throught S1 as leaves throught S2. When external charge is present, it determines the discontinuity in displacement flux. Thus, lines of displacement terminate on external charges. The lines of force, on the other hand, terminate on either external or polarization charges.
4.8.BOUNDARY-VALUE PROBLEMS INVOLVING DIELECTRICS
The fundamental equation that has been develoved in this chapter is
Where is the external charge density. If the dielectrics with which we are concerned are linear, isotropic, and homogeneous, then D = E, winere is a constant charecteristic of the material, and we may write
But the electrostatic field E is derivable from a scalar potential :
So that
Thus, the potential in the dielectric satisfies Poisson’s equation.
4.9.FORCE ON A POINT CHARGE EMBEDED IN A DIELECTRIC
We are determine in a position to detemine the force on a small, spherical, charged conductor embeded in a linear, isotropic dielectric. In the limit in which the conductor is negligibly small from the macroscopic viewpoint, this calculation gives the force on a point charge.
The electric field and surface charge density at a representative point of the conductor surface will be obtained by the boundary-value procedure of the precending section, and the force F may then be obtained from the integral over the surface:
F =
Here E’ stands for the dielectric field at the surface element da minus that part of the field produced by the element itself. In other words,
E’ = E - Es
The force on the conductor becomes
F =
4.10.METHODE OF IMAGES FOR PROBLEMS INVOLVING DIELECTRICS
Furthermore, we surmise the image charge to be located at the same distance d from the interface as the original point charge. Let
and
Then in medium 1
In other words, the image charge q’ is located in medium 2 at position (d,0,0).
For the potential 2 in medium 2, the image charge must be located in medium1, as is also the original charge q, both at position(-d,0,0). We denote the sum of the charges by q”. Then in medium2,
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