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Properties of Conductors

From the discussion, we summarise the following properties of conductors:
  1. The conductivity of a conductor in infinite.
  2. Electric field inside a conductor is zero.
  3. The charge density inside a conductor is zero.
  4. Charges can exist on the surface of the conductor, giving rise to surface charge density.
  5. The electric field at any point on the surface of a conductor is entirely perpendicular to the surface.
  6. A conductor, including its surface, is an equipotential region.

Resistance of a Conductor

If a conductor is maintained at a voltage V then the field inside the conductor is not zero. Description: 22075.png
 Here, the charges will be forced to move, and thus, no static equilibrium is set up. The moving electrons will encounter some damping force, called resistance.

We consider a uniform conductor as shown in Fig. Let
A = Uniform cross section
L = Length of conductor
V = Applied voltage
Description: 40135.png
 Conductor subjected to voltage V

∴ electric field, Description: 24153.png

∴ current density, Description: 33035.png

By Ohm’s law,
Description: 150262.png
Description: 150284.png
∴ resistance, Description: 36886.png

Therefore, we can define resistance as follows.

Joule’s Law

According to Joule’s law, the rate of heat production by a steady current in any part of an electrical circuit is directly proportional to the resistance and to the square of the current (P = I2 R). We will derive the field equation for Joule’s law.

Electromotive Force (emf) and Kirchhoff’s Voltage Law

We know, electric field intensity around any closed path vanishes; i.e., Description: 31475.png
Also, in a conducting medium, the current is, Description: 21364.png
From these two equations, it follows that a purely electrostatic field cannot cause a current to circulate in a closed path (loop). In addition, there must also exist a source of energy to maintain the steady current in a closed loop.

The external source of energy may by non-electrical, such as
  • A chemical reaction (battery)
  • A mechanical drive (dc generator)
  • A light-activated source (solar cell)
  • A temperature-sensitive device (thermocouple)
Difference between Potential (V) and emf (ξ)
  1. Potential field, i.e., electric field generated by static charges, is conservative; but an emf-producing field is nonconservative.
    ∴  Description: 41024.png but, Description: 41017.png
  2. An electric field produced by charges cannot maintain a steady current; but an emf-producing field can maintain a steady current.
  3. Potential (V) is the negative of the line integral of the static field Description: 41010.png while emf (ξ) is the line integral ofDescription: 40994.png Thus, between two points a and b,
    Description: 40986.png
  4. Here, Vab is independent of the path of integration between a and b, but Description: 40978.png is dependent on the path.

Kirchhoff's Current Law

The flux lines in a static electric field region begin and end on electric charge and hence are discontinuous.
The tubes of steady current form a closed circuit on themselves and hence are continuous. For this, the steady current is said to be solenoidal, i.e., no sources or sinks are present.
Thus, as much current must flow into a volume as leaves it. In general, the integral of the normal component of the current density Description: 32852.png over a closed surface S must be zero.
Description: 30798.png Description: 29607.png

This relation for steady current applies to any volume; and may be entirely inside a conducting medium or may be only partially filled with conductors. The conductors may form a network inside the volume or may meet at a point.

An illustration for the conductors is shown in Fig.,
Currents in a closed surface
(I1 - I2 - I3 - I4 - I5) = 0
∴ Description: 41106.png
This is the Kirchhoff’s Current Law (KCL).

Equation of Continuity and Relaxation Time

The differential equation relating the current density Description: 21856.png with the volume charge density Description: 37141.png at a point is known as equation of continuity. This equation is based on the law of conservation of charge which states that charges can neither be created nor destroyed.

Laplace Equation for Conducting Media

For steady current,
Description: 150354.png

∴ Description: 34548.png

Thus, Laplace equation for electrostatic field is the same as that for steady current. This implies that the problems involving distributions of steady currents in conducting media can be handled in the same way as problems involving static field distributions in insulating media.

Principle of Duality

Table: Duality between Description: 23892.png and Description: 22624.png
Sl. No. Description: 23095.png-field Description: 26095.png-field
1. Description: 28920.png Description: 31648.png
2. Description: 31272.png Description: 34825.png
3. Description: 37231.png Description: 26231.png
4. Description: 33935.png Description: 30586.png
5. Description: 22241.png Description: 35146.png
  for dielectrics for conductors

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