7.1.Nature of The Current
The currents we have described thus far in this section are known as conduction currents. These conduction currents represent the drift motion of charge carries in a neutral medium, which is whole, may be and usually is at rest. Liquids and gases may also undergo hydrodynamic motion, and if the medium has a charge density, this hydrodynamic motion will produce currents. Such currents, arising from mass transport of a charged medium, are called convection currents. Convection currents are important to the subject of atmospheric electricity. In fact, the upward convection currents in thunderstorms are sufficient to maintain the normal potential gradient of the atmosphere above the earth. The motion of charged particles in vacuum(such as electrons in a vacuum diode) also constitutes a convection current. A characteristic feature of the convection current is that it is not electrostatically neutral, and its electrostatic charge must usually be taken into account.
7.2.Current Density: Equation of Continuity
if there is more than one kind of charge carrier present, there will be a contribution of the from equation above from each type of the carrier. In general, the current through the area da is
here the summation is over the different carrier types
the quatity in brackets is a vector that has dimensions of current per unit area. This quantity is called the current density and is given the symbol J:
The equation of continuity is
7.3.Ohm’s Law: Conductivity
Thus, the constitutive equation is
J = gE
The constant of proportionality g is called the conductivity.equation above which is called Ohm’s law, is a very good approximation for a large number of the common conducting materials.
The reciprocal of the conductivity is called the resistivity . Thus
The unit of in the mks system is volt-meter per ampere, or simply ohm-meter, where the ohm ( ) is defined by
The unit of conductivity g is -1 m-1 or S/m. One siemens (S) is one reciprocal ohm. (A siemens was formerly called a mho)
7.4.Steady Currents in Continuous Media
There is a very close analogy between an electrostatic system of dielectrics bounded by equiptential surfaces, on the one hand, and a system that conducts a steady current, on the other, this analogy is the subject of the present section.
(steady currents)
Using ohm’s law in combination with equation above, we obtain
Which for a homogeneous medium reduces to
But since x E = 0 for a static field, E is derivable from a scalar potential:
Combination of the last two equations yields
2 = 0
Which is Laplace equation.
Where S is any closed surface that completely surrounds one of the electrodes (except for an insulated metal wire to lead current ontho the electrode so as to maintain its potential constant). But
J = gE
7.5.Approach to Electrostatic Equilibrium
Consider a homogeneous, isotropic medium characterized by conductivity g and permitivity , which has a volume density of prescribed charge 0(x,y,z). If this conducting system is suddenly isolated from applied electric field, it will tend toward the equilibrium situation where there is no excess charge in the interior of the system. According to the equation of continuity,
Which, with the aid of Ohm’s law, becomes
But is related to the sources of the field. In fact, = , so that
The solution to this partial differential equation is, for constant g and ,
And it is seen that the equilibrium state is approached exponentially.
7.6.Resistance Networks and Kirchhoff’s Laws
V = R1I + R2I = (R1 + R2) I
Thus, the equivalent resistance of the combination is
R = R1 + R2 (series conection)
In the parallel conection, the potential difference across each resistor is the same, and the total current through the combination is I = I1 + I2.
And the equivalent resistance R of the combination is obtained from
(parallel combination)
Kirchhoff’s laws may now be stated
I. The algebraic sum of the currents flowing toward a branch point is zero:
II. The algebraic sum of the voltage difference around any loop of the network is zero:
7.7.Microscopic Theory of Conduction
It can be seen immediately that when dv/dt = 0
Vd =
The equation is the steady-state solution for the drift velocity.
if one makes the intial condition v(0) = 0. This equation shows that the local drift velocity approaches its steady value exponentially, like e-t/ , where the relaxation is
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