§5. Conductor in external electrostatic field

5.1. Balance of charges in the conductor

Charge carriers in the conductor are capable of moving under the influence of rather small force and therefore conditions for balance of charges in the conductor have to be done:

1. intensity of a field everywhere inside the conductor is equal to zero

. (5.1.)

It, in turn means that the potential has to be constant, , φ=const.

2) intensity of a field on the surface of the conductor in each point is directed by the normal to the surface

. (5.2.)

Therefore in case of balance of charges the surface of the conductor will be equipotential.

If to provide some charge Q to some body, it will be distributed so that balance conditions will be completed.

Imagine any closed surface which is fully inside. At balance of charges the field in each point inside the conductor is absent therefore the stream of a vector of dielectric shift is equal to zero.

.

Then according to the Gauss's theorem, the total charge inside a surface is also equal to zero (for a surface of any sizes, carried out randomly in the conductor). Following this undle the condition of balance in any place inside the conductor there can not be residual charges: all of them are distributed on a surface density σ. As residual charges aren't present, the removal of substance from some volume which is inside the conductor won't be reflected in an equivalent arrangement of charges.

Thus, the residual charge distributed on the hollow conductor as well as on the solid conductor (with a core), that is on its external surface. Residual charges on the surface of this holowness can't settle down in an equilibrium state. It follows also from the fact that the elementary charges of the same name which form the general charge mutually spurn and seek to settle down at the biggest distance from each other.

Imagine the cylindrical surface formed by normals to the conductor with bases dS, one of which is located inside,another - outside the conductor. The stream of a vector of electric shift through the internal surface is equal to zero, as well inside the conductor,therefore.

E

dS

dS

fig.5.1.

Outside the conductor and in close proximity to it tension is directed by the normal to the surface of the conductor. Therefore for a lateral surface of the cylinder which is outside the cylinder Dn= 0, and for an internal basis D = Dn, that is an external basis is located very close to the basis of the conductor. Respectively, the stream of electric shift is numerically equal to the product DdS, where D - shift size in close proximity to a surface of the conductor. In the cylinder there is a foreign charge.

Inside the cylinder there is a foreign charge.

.

If we use the Gauss's theorem, we will receive:

DdS = σdS,

D = σ.

From this it follows that intensity of a field near the surface of the conductor is numerically equal

, (5.3)

where ε - the dielectric petration of the medium covering this conductor.

We will consider the field created by the charged conductor of the irregular form (fig. 5.2). At long distances from the conductor equipotential surfaces will have configuration characteristic for a dot charge . Approaching to the conductor they become more and more similar to surfaces of this conductor.

Near the peaks of equipotential surfaces will be located more densely as field intensity here is bigger.

Then charge density on the peaks is high enough . It is possible to come to such conclusion if to consider that because of mutual spurning of charges they seek to settle down as far from each other as possible. Near hollows in the conductor equipotential surfaces are located thiner, therefore intensity of a field and area density in these parts will be lower.

fig.5.2. fig.5.3.

In general, charge density at a given value of the potential is defined by curvature of a surface and it increases with the increase in positive curvature (convexity) and decreases with the increase in negative curvature (concavity).

Density of charges on the peak can be especially high and therefore intensity of a field near the peak can be rather big so on peaks there is an ionization of molecules of gas which covers the conductor. Ions of the other sign, than a charge Q are attracted to the conductor and neutralize its charge, and ions of sign Q start moving from the conductor, taking neutral molecules of gas. There appears a movement which is called "an electronic wind". The charge decreases, flows down from the peak and is transferred by "wind". This phenomenon is called as the leakage of a charge from the peak.

5.2. Conductor in external electric field

Properties of solids and their placement in an external electric field are determined by their structure.

Structure of atoms comprises charged particles - electrons and protons. In the normal state the atom is electrically neutral (the number of protons in the nucleus equals the number of electrons moving around it). The electrons are kept at the nucleus by electrical attraction to the nucleus, but the electrons in metals can be removed out of "their" atoms and move freely until they are trapped by other atoms.

The electrons that have lost touch with its atom are called free. Their movement is chaotic and depends on the temperature of the external enviroment. In metal conductors concentration of free electrons reaches . This movement of electrons is similar to thermal motion of free gas and a combination of free electrons is calledan electron gas.

If a metallic conductor is placed in an electric field, the ordered motion is imposed on the chaotic motion of the electrons is the direction opposite to the field strength. This movement is called the drift .

The drift will be toward the surface AB (fig. 5.4), so that an excess positive charge will appear on CD. The phenomenon of charge redistribution due to the action of an external electric field is called electrostatic induction, and the charges that arise on the surface of the conductor are called given or induced.

Charges will be redistributed as long as Е₀ is not equal to the opposite direction of tension E induced charges.

A C

B D

E0

fig.5.4.

The intensity of the resulting field will be zero and orderly movement of charges stop - come equilibrium. If the conductor charges in equilibrium, the field potential does not change. Thus, in the case of equilibrium surface of the conductor is equipotential.

5.3. Electric capacity intensity. Capacitors capacitance

Experimentally, it was found that when the shape and size of the conductor and the environment does not change, with increasing charge the potential of conductor increases proportionally.

,

,

where C - electricity intensity conductor.

Electric capacity of a conductor or a system of conductors is a physical property that describes the ability of a conductor to accumulate electrical charges. In general, electric capasity depends on the environment and the location of surrounding bodies, but is independent from the charge and the capacity of a conductor.

1 F – the capacity of the conductor, which changes the potential at 1B by changing the charge on 1K.

The capacity of a spherical conductor can be determined from the formula:

,

As it was mentioned, in general capacity depends on the medium in which the conductor is placed. A system of conductors, the capacity of which does not depend on the location of surrounding objects is called the capacitor - a system of two conductors (plates) separated by a insulator, the thickness of which is small compared to the size of the plates. Capacitor plates are arranged so that the field created by charges is concentrated in the space between the plates.

Capacitance is determined by the geometry and dielectric properties of the medium that fills the space between the plates.

In the course of charging, a charge equal in size and opposite in sign appears on the plates. The potential of the field between the plates is proportional to the charge of the capacitor plates.

The forms of capacitors are followings:

1) flat capacitor

 

(5.4)

where d is the distance between the plates, S - area of each capacitor plates.

d

fig.5.4

2) cylindrical capacitor

 

(5.5)

where R, r - radii of coaxial cylinders, L - length of generators.

R

r

L

fig.5.5

                

3) spherical capacitor

 

(5.6)

where R, r - radius of the sphere.

     

r

R

fig.5.7

If the capacitor has multiple dielectrics, its capacity is determined by the formula:

  .

d₁

d₂

d₃

fig.5.8