In the previous post we derived that current density is j = nqv_{d}
where v_{d} is the drift velocity.

Current density in general depend on electric field and for metals current density is nearly proportional to the electric field. (Results can be derived using theory of metallic conduction.)

Thus for metals ratio of E and j is constant and for a particular material its resistivity ρ is defined as the ratio of magnitude of electric field to current density,
ρ = E/j
This relationship is known as Ohm's law discovered by german physicist Georg Simon Ohm (1787-1854) in 1826.

Greater would be the resistivity of a given material greater field would be required to establish a given current density in the material or we can say that smaller would be the current density for a given field.

Unit of resistivity is Ωm (ohm. meter).

Materials having zero resistivity are known as perfect conductors and those having infinite resistivities are known as perfect insulators. Real materials lie between these two limits.

Metals and alloys are materials having lowest resistivities and are good conductors of electricity.

Insulators have resistivities many times (of the order of 10^{22}) greater then that of metals.

Reciprocal of resistivity is conductivity. Unit of conductivity is (Ωm)^{-1}.

Metals or good conductors of electricity have conductivity greater than that of insulators.

Semiconductors are those materials which have resistivities intermediate between those of metals and insulators.

(B) Resistivity and temperature

Resistivity of a conductor depends on a number of factors and temperature of the metal is one such factor. As the temperature of the conductor is increased its resistivity also increases.

For small variations in temperature resistivity of materials is given by the relation
ρ(T) = ρ(T_{0})[ 1 +α(T-T_{0})]
where, ρ(T) and ρ(T_{0}) are resistivities at temperature T and T_{0} respectively and α is constant for a given material which also depends on temperature to a small extent. This constant α is known as temperature coefficent of resistivity.

(C) Resistance

We already know that for a conducror relation between electric field E and current density is given as
E = ρj
where ρ is a constant independent of E.

When we study electric circuits we are more interested in the total current in a conductor rather then current density j and more interested in knowing the potential difference between the ends of the conductor than in Electric field becaude current and potential difference are easier to measure then j and E.

Consider a conducting wire of length l and uniform crossectional area A. If V is the potential difference between both the ends of the wire then electric field inside the conductor would be
E = V/l
If i is the current flowing inside the wire then current density is given by
j = i/A
putting these values in Ohm's law ρ = E/j we get
V = ρi (l/A)
or , V=Ri
where, R=ρ(l/A)
which is known as resistance of a given conductor.

Unit of resistance is ohm or volt per ampere.

Thus how much current will flow in a wire not only depends on the potential difference between two ends of the wire but also on the resistance offered by the conductor to the flow of electric charge.

From the above discussion we can easily conclude that The resistance of a wire depends both on the thickness and length of the wire and also on its resistivity.

Thick wires have less resistance then thin ones and longer wires have more resistance then shorter ones.

Since the resistivity of a marerial varies with temperature, the resistance of any particular conductor also varies with temperature. For temperature ranges that are not too great, this variation is approximately a linear relationship, analogous to the one we learned for resistivity
R(T) = R(T_{0})[1 + α(T - T_{0})]
In this equation. R (T) is the resistance at temperature T and R(T_{0}) is the resistance at temperature T_{0}. The temperature coefficient of resistance α is the same constant that appears in case of resistivity.

In the next post we'll do some worked examples related to this topic

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