Temperature Dependence of Resistance

Devices such as batteries, cells, etc. are essential for maintaining a potential difference across the circuit and are referred to as voltage sources. When a voltage source is connected across a conductor, it creates an electric field which causes the charges to move and this causes current. The values of the current that are generated strictly depends on the characteristic of the material. Any material opposes the flow of electric charge which is called resistance. It is developed due to resistivity which is a property of the material. Let’s see how these properties change with temperature when it is varied. 

Resistance 

Water flowing through a pipe encounters resistance to its flow if the current is considered as water and the pipe is considered to be the conductor through which the current is flowing. The same analogy can be used to deduce the obstruction of flow in the case of current too. This obstruction of flow against the current is termed Resistance. The resistance in a wire experiencing a current I and voltage V is defined as, 

R = V/I

Here, R denotes the resistance of the wire. Its unit is “ohms” which is denoted by Ω

Notice in the relation, the resistance depends inversely on the current flowing through the wire. Thus, the more the resistance of the wire, the less is the current flowing through it, which can also be deduced from the intuitive definition of the resistance. 

Resistance of a Wire

Picking the previous example of flowing water, one can draw the similarities between these two very different physical processes. The resistance offered to the water flowing through the pipe majorly depends on the length of the pipe and its cross-sections. In a similar way, the resistance offered by a conductor depends on the conductor’s size, shape, and material. The figure below shows a conductor which the length l, cross-sectional area A and resistivity (rho). 

The resistance of the conductor is given by, 

R = ρl/A

Notice that, the resistance is proportional to the length of the conductor but inversely proportional to the cross-sectional area of the conductor. 

Resistivity 

When a voltage is applied to a conductor, the electric field E is created inside it which pushes the charges to move. The current density that gets developed depends on the material and the electric field that is created. This density can be very complex but under reasonable assumptions, including assuming the metals at room temperature. This relation can be modeled using, 

J = σE

Here, σ is the electrical conductivity.

The unit of resistivity is given by ohm-meter(Ω-m). 

Temperature Dependence of Resistivity

As mentioned in the formula above, the resistance of a material. depends on its resistivity, shape, and size. Resistance of a material depends on how the shape changes with temperature and resistivity variations with temperature. Resistivity behaves differently with temperature for different materials. Typically, conductors have low resistivity while insulators have high resistivity. Resistivity varies in a different manner for different materials. In general, for metallic conductors,

The resistivity of metallic conductors within a limited range of temperature is given by the following equation: 

ρ_T = ρ₀[1 + a(T – T₀)]

Here, 

ρ_T = Resistivity at temperature T, 

ρ₀ = Resistivity at temperature T₀, 

a = temperature coefficient of resistivity. 

Resistivity varies differently with different materials. For example, materials like Manganin, Nichrome are less likely to change their resistivity with temperature. In the case of semiconductors, their resistivity decreases with temperature. 

Sample Problems 

Question 1: A battery of 20 Volts connected to a conductor induces a current of 20mA in the conductor. Find the resistance of the conductor. 

Answer: 

The resistance of conductor is given by the relation, 

R  = V/I 

Given: 

V = 20V 

I = 20mA = 0.02 A 

Plugging in the values inside the relation, 

R = V/I 

⇒ R = (20)/(0.02) 

⇒ R= 1000 Ohms. 

Question 2: A battery of 10 Volts connected to a material induces a current of 0mA in the conductor. Find the resistance of the conductor. 

Answer: 

The resistance of conductor is given by the relation, 

R  = V/I 

Given: 

V = 10V 

I = 0mA = 0 A 

Plugging in the values inside the relation, 

R = V/I 

⇒ R = (10)/(0) 

The resistance is closes to infinity, which means the material is an insulator. 

Question 3: A cylindrical conductor of length 0.1m and cross-sectional area 0.01 m². The resistivity of the material is 1 x 10^-8 ohm-m. Find the resistance of the material. 

Answer: 

The resistance of a conductor is given by, 

R = 

Given 

l = 0.1m 

A = 0.01 m²

ρ = 1 x 10^-8 

Plugging the values in the relation given above,

 R = 

⇒ R = 

⇒ R = 10^-7

Question 4: At temperature T₀ the resistivity of a metallic conductor is 15.4 ohm-m. Let’s say the temperature is increased by 50K and the temperature coefficient of resistivity is given by 0.0045. Find the new resistivity. 

Answer: 

The resistivity of metallic conductors is given by the following equation: 

ρ_T = ρ₀[ 1 + a(T – T₀)]

Here, 

ρ_T =?

ρ₀ = 15.4 nOhm-n 

a = 0.0045 

ρ_T = ρ₀[ 1 + a(T – T₀)]

⇒ ρ_T = 15.4[ 1 + (0.0045)(50)]

⇒ ρ_T = 15.4[1.225] 

⇒ ρ_T = 18.86

Question 5: At temperature T₀ the resistivity of a metallic conductor is 30.8 nohm-m. Let’s say the temperature is increased by 100K and the temperature coefficient of resistivity is given by 0.009. Find the new resistivity. 

Answer: 

The resistivity of metallic conductors is given by the following equation: 

ρ_T = ρ₀[ 1 + a(T – T₀)]

Here, 

ρ_T =?

ρ₀ = 30.8 nOhm-n 

a = 0.009

ρ_T = ρ₀[ 1 + a(T – T₀)]

⇒ ρ_T = 30.8[ 1 + (0.009)(100)]

⇒ ρ_T = 30.8[1.9] 

⇒ ρ_T = 58.52