Problem 14 - Physics

Question

A coil of wire has a resistance of \(38.0 \Omega\) at \(25^{\circ} \mathrm{C}\) and \(43.7 \Omega\) at \(55^{\circ} \mathrm{C}\). What is the temperature coefficient of resistivity?

Step-by-Step Solution

Answer

The temperature coefficient of resistivity is approximately \( 5 \times 10^{-3} \, \mathrm{per} \, ^{\circ} \mathrm{C} \).

Step 1: Identify the Known Values

We know the resistance of the coil at two different temperatures: \( R_1 = 38.0 \, \Omega \) at \( T_1 = 25^{\circ} \mathrm{C} \) and \( R_2 = 43.7 \, \Omega \) at \( T_2 = 55^{\circ} \mathrm{C} \).

Step 2: Understand the Formula

The formula for resistance change with temperature is \( R = R_0 (1 + \alpha(T - T_0)) \), where \( \alpha \) is the temperature coefficient of resistivity. We can express the change as \( R_2 = R_1 (1 + \alpha(T_2 - T_1)) \).

Step 3: Rearrange the Formula

Rearrange the formula to solve for \( \alpha \). This gives: \( \alpha = \frac{R_2 - R_1}{R_1 (T_2 - T_1)} \).

Step 4: Substitute the Values

Substitute the known values into the formula: \( \alpha = \frac{43.7 - 38.0}{38.0 \times (55 - 25)} \).

Step 5: Calculate \(\alpha\)

Calculate \( \alpha \) using the substituted values:\[ \alpha = \frac{5.7}{38.0 \times 30} \approx \frac{5.7}{1140} \approx 0.005 \], or approximately \( 5 \times 10^{-3} \, \mathrm{per} \, ^{\circ} \mathrm{C} \).

Key Concepts

Resistance

In electronics, resistance is a fundamental concept that refers to the opposition that a material offers to the flow of electric current. Imagine trying to push water through a hose; the narrower the hose, the harder it is to push water through it, similar to how resistance works in an electrical circuit.
Resistors, devices designed to provide resistance, are used to control the flow of current and to divide voltages in a circuit. Ohm's Law, which states that the voltage across a resistor is directly proportional to the current flowing through it, is pivotal in understanding resistance. It helps us express this relationship mathematically:

• Where resistance (\( R \)) is defined by the equation \( R = V/I \).

The unit of resistance is the ohm (\( \Omega \)). High resistance means low current flow, while low resistance means more current flow. Materials like copper that allow current to pass easily have low resistance. Materials like rubber have high resistance.

Temperature Effects on Resistance

Temperature can significantly impact electrical resistance. Most metals increase in resistance as the temperature rises. This is because increased temperature causes atoms in the metal to vibrate more, which makes it harder for electrons to pass through, much like a concert crowd blocking passage.
The formula for understanding how resistance changes with temperature is given by:

• \( R = R_0 (1 + \alpha(T - T_0)) \)

Where \( R \) is the resistance at the new temperature, \( R_0 \) is the original resistance, and \( \alpha \) is the temperature coefficient of resistivity.
This coefficient \( (\alpha) \) shifts depending on the material and shows how much resistance changes per degree Celsius. This was illustrated in our initial exercise with a coil of wire, demonstrating that resistance increases from \( 38.0 \, \Omega \) to \( 43.7 \, \Omega \) when the temperature rises from \( 25^{\circ} \mathrm{C} \) to \( 55^{\circ} \mathrm{C} \).

Electrical Conductivity

Electrical conductivity is the ease with which an electric charge or heat can pass through a material. Imagine a superhighway on which electrons can speed along their path with little to no obstacles. Materials with high conductivity allow charges to move freely, while those with low conductivity resist the flow.
Conductivity is inversely related to resistivity — the higher the conductivity, the lower the resistivity, and vice-versa. This is captured in the relationship:

• \( \sigma = \frac{1}{\rho} \)

Where \( \sigma \) is the conductivity and \( \rho \) is the resistivity.
Metals like silver and copper are highly conductive, thus widely used in electrical wiring. Conductivity is critically important in designing materials and devices for different electrical applications, considering how efficiently they can transport current.
This concept helps us grasp why materials act differently when conducting electricity and is paramount not only in electronics but in everyday technologies that depend on electrical energy utilization.