Determination of a relationship between Co-efficient of Linear Thermal Expansion of Solid with respect to the Resistivity of the material

Being an inquirer and a creative thinker, I always tried to decipher the scientific reason behind every observation in the world. This inquisitive nature has landed me to clear the concepts of physics and applications of physics and real-life situations. Though most of the curiosities are answered, however, few are left unanswered. The reason behind this exploration was one of such an inquiry. While travelling from my brother’s place to my hometown, due to unavailability of flight ticket, I travelled via train where this curiosity came to my mind. I observed that there are gaps at the junction of two consecutive rails. After reaching my place, I started research on it. Recently, in my curriculum, I learnt that when the temperature increases, the solid metal usually expands due to the phenomenon of thermal expansion of solid. Learning the above phenomenon, I tried to relate the observation with different instances where this case will be prevalent. Hence, one thing which come to my mind was electricity. What is the effect of extension of solid on current flow in conductor? Does it affect the other microscopic properties that controls the flow of charge?

 

To decipher the answer, I started my research on it. I read quite a few research papers and journals on thermal expansion of solid and understood that the temperature difference is the factor that affects the extension and not the exact value of temperature. However, after taking a few video lectures and tutorials, I was able to grasp confidence and knowledge on the domain of physics but couldn’t find any relationship between expansion of solid due to change in temperature and the factors controlling current flow. This inquiry resulted me landing in this research question stated below.

How does the coefficient of thermal expansion of a wire (expressed in ℃^-1) due to heating by continuous flow of current through the wire depend on the resistivity of the wire, expressed in ohm. m (copper - 0.0171, aluminum - 2.8 × 10^-8, nichrome - 1.1 × 10^-6, iron - 1.0×10^-7 and tungsten - 4.9 × 10^-8), determined using length versus temperature scattered graph?

With an increase in temperature of any object, the length of solid increases because of an increment in the kinetic energy of the molecules of the object. The formula of increase in length is expressed as:

 

∆L =  L₀ × α × ∆T

 

= > α = \(\frac{∆L}{L_0×∆T}\)..........(5)

 

Here, ∆L indicates the extension in length of object,  indicates the coefficient of thermal expansion, and ∆T indicates the difference in temperature.

 

Co-efficient of thermal expansion can be defined as the increase in length of a solid object of unit length for an increase in temperature of unity.

Resistivity of conductor (ρ) is the resistance offered by a unit length and a unit area of cross-section of a conductor. It is expressed in ohm. m.

 

Resistance offered by a conductor depends on certain factors which are shown below:

 

• Length of the conductor

Length is directly proportional to the resistance offered by the conductor.

 

R∝L………(equation-1)

 

• Cross-sectional area of the conductor

Cross-sectional area is inversely proportional to the resistance offered by the conductor.

 

R∝ \(\frac{1}{A}\)………(equation-2)

 

Combining equation (1) and (2), it has been experimentally found that:

 

R = ρ × \(\frac{L}{A}\) ………(equation-3)

 

R = Resistance offered by the conductor

 

L = Length of the conductor

 

A = Area of cross-section of the conductor

 

ρ = Resistivity of the conductor

A conductor dissipates heat current flows through it. It is stated that, amount of heat liberated is directly proportional to the square of current, resistance and time (in seconds) through which the current is flowing through the conductor.

 

H = I² Rt

 

H = Heat dissipated by the conductor

 

I = Current flowing through the conductor

 

R = Resistance of the conductor

 

t = Time through which current is flowing through the conductor

The molecular arrangement of solid or the properties of a solids are stated below:

• Solids have a definite shape.
• Solids have a definite volume.
• The intermolecular force of attraction in solids is very high. As a result, molecules of solids are very closely packed.
• The intermolecular space in solid is very less and quite compared to negligible space.
• Solids cannot be compressed.
• As the internal energy (kinetic energy) of the solid molecules cannot overcome the intermolecular force of attraction of solids, thus, the solid molecules vibrate at their own position.

It predicts the strength of a derived relationship between a dependent variable and the independent variable. The maximum value of the coefficient is ±1 which indicates maximum strength and the minimum value of the coefficient is 0 which signifies no correlation between the two variables. A negative value indicates that the relationship between the independent and the dependent variable is decreasing and, a positive value signifies that the relationship between the independent and the dependent variable is direct in nature. The formula of Pearson’s correlation coefficient for a linear trend is shown below:

 

R = \(\frac{\sum(x\ -\ \bar{x})(y\ -\ \bar{y})}{\sqrt{\sum(x\ -\ \bar{x})^2 \ ×\ \sum(y\ -\ \bar{y})^2}}\)

 

R = Pearson's Correlation Coefficient

 

x = each value of independent variable

 

\(\bar x\) = mean value of independent variable

 

y = value of dependent variable corresponding to the value of x

 

\(\bar y\) = mean value of dependent variable

In this process, a very simple circuit was formed with a voltage source and a limiting resistance connected in series in the circuit. Each type of wire was used as the connecting wire in the circuit for different trials. When the voltage source was turned on, due to flow of current through the circuit, some amount of heat was dissipated by the resistor as well as the wire. As the exploration is mainly about the resistivity and thermal expansion of wire, the temperature of the wire was monitored using a temperature probe. At an interval of 10.00 ℃ from the room temperature of 20.00℃, the battery was turned off and the expansion of length of wire was measured using a meter scale and a vernier calipers. The length of the wire was measured till a temperature of 60.00℃. Using the length of extension of wire at different temperature, the coefficient of thermal expansion of different materials were calculated and its relationship with the resistivity of wire was studied.

In a research article titled as – ‘Thermal expansion of solids’ in Moscow Izdatel Nauka (1974), by Novikova, So I, it was concluded that the coefficient of thermal expansion is a function of time and not a constant parameter. As a result, it could be assumed that for different materials, if the experimental duration varies, then the coefficient of thermal expansion would behave differently.

It was assumed that with an increase in resistivity of material, the coefficient of linear thermal expansion of the material would increase. The reason behind this hypothesis could be explained in two steps. Firstly, with an increase in temperature of a conductor, the resistance of the conductor increases. This was because, with an increase in temperature, the molecules of conductor (solid) vibrate with an intensified rhythm which, consequently, has increased the collision between the molecules and the free moving electrons. Due to increased collision, the flow of electrons was interrupted which resulted in an increase in resistance of the conductor. As the resistance of the conductor was increased due to microscopic property, it was evident from the explanation that the increase in resistance was due to an increase in resistivity of the material.

 

Secondly, with an increase in coefficient of thermal expansion, it could be assumed that the intensity of vibration of molecules of solid would increase due to which the length of the solid eventually increase. As the intensity of vibration of molecules would increase with an increase in coefficient of linear expansion, the resistivity would increase.

The resistivity of material, measured in ohm. m was chosen to be the independent variable of the exploration. Five materials were considered in this exploration to study the variation of the dependent variable with respect to the change in resistivity of materials. The five materials chosen for exploration are as follows: 1. Copper, 2. Aluminum, 3. Nichrome, 4. Iron and 5. Tungsten. These five materials are chosen because of the exploration methodology. The dependent variable is obtained from a length of metal wires (solid) of different resistivity was measured against temperature of the material using a scatter plot. In the process, as resistivity of the material is the independent variable, five materials are chosen in such a way that those are used as materials of wire and conduct electricity. Moreover, in the experimental procedure, the principle of Joule’s Law of Heating is used to raise the temperature of the wires. As a result, it is very important for the materials to conduct electricity through them. As the materials are chosen in accordance with the requirement of flow of current through it, the interval of resistivity between any two material is not constant. Rather, the intervals are quite irregular and invariably large.

The coefficient of linear thermal expansion was considered to be the dependent variable of the exploration. It was measured in the units of ℃^-1. For each wire (material), the coefficient of linear thermal expansion was calculated using the formula of change in length of a solid material due to change in temperature (refer to section 3.1), from the experimentally observed value of the exploration.

There were no controlled variables in this exploration. This was because, both the dependent and the independent variable are intensive properties of matter. However, few physical parameters were considered to constant throughout the experimental procedure to maintain a uniformity in the exploration.

• 300 cm wire of diameter of cross section of 20 AWG of each type was taken and intervals at 100.00 cm using a vernier calipers and the meter scale was marked using a marker pen.
• At the markings, the wires were cut using a wire cutter and three pieces of each type of wire of 100.00 cm each was obtained.
• The room temperature was measured using a room thermometer.

• Rubber gloves and leather shoe were worn throughout the experimental procedure.
• A battery (voltage source) of 5 V and a resistor of 100 Ω was taken.
• The positive terminal of battery was connected with one terminal of the resistor using a crocodile-to-crocodile probe.
• The negative terminal of the resistor connected with a piece of wire of one type obtained in the pre-experimental arrangement using a crocodile-to-crocodile probe.
• The other end of the wire is connected with the negative terminal of the battery using a crocodile-to-crocodile probe.
• The room temperature was measured using a room thermometer.
• The battery was turned on and the temperature of wire is monitored using a temperature probe.
• At the specific temperatures of 30.00℃, 40.00℃, 50.00℃, and 60.00℃, the battery was turned off and the wire was removed from the circuit. The length of the wire was measured using a vernier calipers and a meter scale and noted. After that, the wire was placed again into the circuit in the same arrangement.
• Once the previous step was undertaken, the wire was replaced by another piece of wire of same type and the same process was repeated two more times for two other pieces of wire of same type.
• The same process was then repeated for the other types of wires.

• Rubber gloves and leather shoes were worn throughout the experimental process to avoid getting electric shock.
• Rubber gloves were also worn throughout the experimental process to avoid any burn on skin and avoid touching the high temperature wire during measuring of length of the wire at different temperatures.