1. Chapter five conduction
Certain materials like Cu, Ag, Au, etc. have free electron of free charges, when an electric field is applied to these materials, the free charges being to move in the direction of the electric field. This process of motion of electrical charges is called conduction. The particle like electron, positive or negative ions or the hole which transfer the electrical charge from one place to other place are called the charge carriers. In any conductor, the charge carriers at a given temperature move with random velocity. When an electric field is applied, these randomly moving charge carriers being to move slowly in the direction of the electric field . The velocity of the group of randomly moving charge carriers is called drift velocity.

2. 5-1 Electric current:
Let us consider a conductor ABCD as shown, let the number of charge carriers per unit volume be N, let the charge on each carriers be q , let us apply an electric field so that the drift is . Consider a small area so that normal to thus area makes an angle with . We want to find the number of charge carriers passing through this area per unit time. Consider a cylinder abcd as shown in diagram, all the charge carriers in the cylinder will pass through da in unit time. Therefore the charge passing through the area in unit time in a direction normal to the field
Where and are the density, charge and velocity of ith specie. is called the electric current, therefore the electric current is the charge passing through any area normal to the electric field per second.
vector quantity

3. 5-2 Current density: The current density J in any conductor is defined as the current per unit area normal to the electric field from eq. (1) we can write Therefore the relation between the current I and the current density J is 5-3 Equation of continuity: Let us suppose that we have a conductor ABCD and an electric field E is applied as shown in the diagram. Let the current density be J. consider a closed surface S of volume V. then the current entering this closed surface is If the charge density in the volume is , then the charge in this volume ,

4. therefore the current going out because is a function of position and time and we need only charge of with time, therefore the current going out
Now the current going out = current going in the closed surface Or
Or (4)
Eq. (4) is called equation of continuity.

5. 5-4 Ohm’s law:
From the definition of J, we know that J depends on the applied electric field
Or (5)
Where g is the conductivity of the material and is called resistivity of the material
Eq. (5) becomes
Eq. (5) and (6) are two forms of Ohm’s law. Now we know that the potential difference across a length is given by
From eq. (6) and (7) we get
Or because
R is the resistance of length dl. Eq. (8) is the well-known form of Ohm’s law .

6. 5-5 Steady current in a conductor:
Let us consider a conductor having a constant current I. in this problem we can apply equation of continuity. Therefore if J is the current density and ρ is the charge carries density then,
Since the current is steady, therefore eq.(9) becomes
Using Ohm’s law eq. (10) becomes
Or where u is the potential difference
Eq. (11) is the Laplace’s equation. This means that equation of continuity gives Laplace’s equation when current is steady or Laplace’s equation can be used to solve the problem of steady current.

7. Approach to a steady current I:
Let us consider a conductor AB of conductivity g, connected as shown in the diagram, when the key k is pressed, the current in the conductor rises from zero to steady value I.
From eq. of continuity, we can write ∂ρ/∂t + Div J=0
Or ∂ρ/∂t + Div gE= (12) , J = gE
Now if ρ is charge density at any time t , then
Div E=ρ/ϵ (13)
Where ϵ is the permittivity of the conductor. From eq. (13) we have ∂ρ/∂t + g/ϵ ρ=0
∂ρ/∂t=-g/ϵ ρ ∫∂ρ/ρ=∫(-g/ϵ dt)
lnρ = -g/ϵ t + c (14)