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### Electric resistance and Resistivity

(A) Resistivity

• In the previous post we derived that current density is
j = nqvd
where vd 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 1022) 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(T0)[ 1 +α(T-T0)]
where, ρ(T) and ρ(T0) are resistivities at temperature T and T0 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(T0)[1 + α(T - T0)]
In this equation. R (T) is the resistance at temperature T and R(T0) is the resistance at temperature T0. 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