Electrical Resistivity and Conductivity

Resistance

Electric current that flows in a circuit is as similar as the water flowing through a river. In a river rocks, branches and other particles resists the flow of water. Likewise in a circuit there are elements which may resist the flow of electrons. This property of resisting the flow of electrons or the current is called the Resistance.

The unit of resistance is ohm. One ohm is equal to volt per ampere.

From ohm’s law we have seen that

R =

· where V is the voltage and I is the current.

Resistance (R):- Resistance of a conductor is defined as the ratio between potential differences between the two ends of the conductor to the current flowing through it.

R= V/I

Relation between resistivity(ρ) and relaxation time (τ):-

ρ = m /ne² $\tau$

Resistivity

Resistivity is the resistance per unit length and cross sectional area. It is the property of the material that opposes the flow of charge or the flow of electric current.

The unit of resistivity is ohm meter.

We know that

R = ρ .

Thus we can derive the expression for resistivity from this formula.

ρ = R

Where R is the resistance in ohms

A is the area of cross section in square meters

L is the length in meters

When the values of L, the length and A, the area is equal to one, we can say that the resistivity is equal to the resistance. So resistivity can be defined as the specific resistance of a material. When we have a thick wire, the resistance decreases. The resistance increases when the wire is thin as the area of cross – section is less. When the length of wire increases, the resistance also increases. When the length of the wire decreases, the resistance decreases as the length is less.

Resistivity

A material with high resistivity means it has got high resistance and will resist the flow of electrons. A material with low resistivity means it has low resistance and thus the electrons flow smoothly through the material. Copper and aluminum has low resistivity. Good conductors have less resistivity. Insulators have high resistivity. The resistivity of semi – conductors lies between conductors and insulators. Gold is a good conductor of electricity and so it has low resistivity. Glass is a good insulator which does not allow the flow of electrons. Hence it has high resistivity. Silicon is a semi -conductor and so it allows partial movements of electrons. Resistivity of Silicon comes between glass and gold. The resistivity for perfect conductors is zero and the resistivity for perfect insulators is infinite.

Variation of Resistivity with respect to temperature

The resistivity of materials is based upon their atomic structure. So we can change the resistivity of materials by changing the temperature. We know that the valence electrons are loosely bound to the nucleus. In metals at normal temperature, although the electrons collide with the metal atoms the free electrons move freely. So as there is some resistance in the metal the current still flows.

When the temperature is increased, the metal atoms starts vibrating and then occurs a random motion. Thus the free electrons could move very slowly as compared to the case of normal temperature. When the temperature increases the hindrance increases and thus the resistivity also increases. When the atoms start vibrating with more amplitude, the collision becomes more frequent. Thus the drift velocity also decreases and then the current starts decreasing.

When temperature increases on metals resistivity also increases

In non – metals the electrons are tightly bound to the nucleus. When the temperature applied is too high, the electrons tend to be loose themselves from the atoms and come out of the atom for the conduction. So the conductivity increases. When the conductivity increases, the resistivity decreases and thus the flow of current increases.

When the current passes through a material, it heats up. When the temperature of the material changes, the resistance also changes. In case of most of the resistors the effect will be too small. But for some other resistors we can see the effect as very large. The resistors with large effect can be used as a temperature sensor. With a voltage of known value placed across the resistor and measuring the current we will get the resistance of the material. Thus we will get the temperature of the material to which the resistor is connected. Hence it can be used as a temperature sensor.

Variation of resistance with temperature:-

Temperature coefficient of resistance (α) is defined as change in resistance of the conductor per unit resistance per degree centigrade rise of temperature.

R_t=R₀[1+α(T-T₀)]

α = R_t- R₀/R₀(T-T₀)

Here, R_t,R₀ is the resistance of the conductor at tº C and 0º C respectively.

Resistivity of material (ρ):- ρ = RA/l ,

Here R is the resisteance of the conductor, A is cross sectional area of conductor and l is the length of the conductor

Temperature co–efficient of resistivity

ρ_t = ρ₀ [1 + α (T – T⁰)] is the equation for the temperature co- efficient of resistivity.

ρ₀ is the resistivity at a standard temperature, ρ_t is the resistivity at t⁰ C, T₀ is the reference temperature, α is the temperature co–efficient of resistivity. For metallic conductors α, the temperature co –efficient of resistivity is a positive value. The value of α is negative for semiconductors and insulators.

Conductivity

Conductivity is the reciprocal of resistivity. Electrical conductivity is the ability of the material to allow the movement of free electrons. Thus it allows to conduct electricity.

σ =