Resistivity of Various Materials

Resistance

We know that 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 = V / I, Where V is the voltage and I is the current.

Resistors are used to resist or control the flow of electrons by the conductive material. They do not provide any power to the circuit. They may reduce the voltage and current passing through the circuit. Hence resistors are passive devices. Most of the resistors are made up of carbon, metal or metal oxide film.

Resistors

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 L / A. Thus we can derive the expression for resistivity from this formula. ρ = R A / L, where R is the resistance in ohms, A is the area of cross section in square meters and 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.

Conductors, Insulators and Semiconductors

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.

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.
 

Ohm’s Law and Resistivity

We know that ohm’s voltage is equal to current times resistance, which is V = I R. It says that the voltage is directly proportional to the current when the resistance is constant. The resistance of a material does not change when the temperature remains constant.

We know that the drift velocity, V_d  = - e Eԏ /m – equation 1, where e is the charge on an electron, E is the electric field, ԏ is the average time taken between each collision or the relaxation time of electrons and m is the mass of the electron. When an external electric field is applied across the conductor, the electrons drift to the positive or the higher potential end of the conductor at a certain velocity. This net velocity is called the Drift Velocity.

Also E = - V/L where V is the potential difference and L is the length. Electric field is the negative gradient of electric potential.

Substituting the value of E in the first equation we get V_d  = e V ԏ / m L 

From this equation we get V = V_d m L / e ԏ - equation 2. 

Current I = A n e V_d, where A is the area of the cross section of the wire, n is the charge density, e is the charge on current carriers or electrons, V_d is the drift velocity.

From the equation we get V_d = I / A n e.

Substituting this value of V_d  in equation 2 we get V = I / A n e *( m L / e ԏ)

Re arranging the equation we get V = (m L / A n e² ԏ) I – equation 3

We can say that (m / n e² ԏ) * L /A = R, the resistance and also we know that R = ρ * L / A

So m / n e² ԏ = ρ

Thus substituting the values in the equation 3 we get, V= ρ * L / A * I

= R I

Thus V = I R. Finally we derived the ohm’s law and it is proved.
 

Resistivity on Electric Field and Current Density

The resistivity also depends on the magnitude of the electric field and current density. The formula is ρ = E / J, where E is the electric field and J is the current density. Electric field is measured in volts per meter. The current density is measured in amperes per meter square. Current density is the flow of electric charge per unit area of cross section. We can also see that if the resistivity of a material is high, then the electric field applied to the material to develop a given current density will also be high.
 

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 = 1 / ρ, where σ is the conductivity and ρ is the resistivity. The unit of conductivity is ohm ^-1 m ^-1      

σ = 1 / ρ = n e² ԏ / m, where n is the charge density, e is the charge on current carriers, ԏ is the relaxation time of electrons and m is the mass of the electron.

The table given below provides the details of resistivity, conductivity and temperature co- efficient of various materials.

Summary

• The resistance is the property that resists the flow of electrons or the current in a circuit. Ohm is the unit of resistance. Resistors resists the flow of electrons by the conductive material. The resistors are passive devices as they do not provide any power to the circuit. It only reduce the voltage and current passing through the circuit.

• R = ρ L / A. Thus we can derive the expression for resistivity from this formula. ρ = R A/L, where R is the resistance in ohms, A is the area of cross section in square meters and L is the length in meters. Resistivity is also known as specific resistance.

• Good conductors like gold have less resistivity. Insulators like glass have high resistivity. The resistivity of semi – conductors like silicon lies between conductors and insulators that is between gold and glass.

• Based upon their atomic structure the resistivity of materials tend to change.

• When the temperature is increased in metals, the metal atoms vibrates, frequent collision occurs. Then the drift velocity also decreases and thus the resistivity increases. So the current starts decreasing.

• When the temperature is increased for non - metals, the resistivity decreases and so the current flow increases.

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

• Ohm’s law which is V = I R is verified using the term resistivity.

• ρ = E / J. This indicates that the resistivity depends on the magnitude of the electric field (E) and current density (J).

• The reciprocal of resistivity is called as conductivity. σ = 1 / ρ, σ is the conductivity.

Watch this Video for more reference
 

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Resistivity of Various Materials