Question

The temperature co-efficient of resistance of a wire is \(0.00125^{\circ} \mathrm{k}^{-1}\). Its resistance is \(1 \Omega\) at \(300 \mathrm{~K}\). Its resistance will be \(2 \Omega\) at. (A) \(1400 \mathrm{~K}\) (B) \(1200 \mathrm{~K}\) (C) \(1000 \mathrm{~K}\) (D) \(800 \mathrm{~K}\)

Short answer

The correct answer is (B) 1200 K, which is the closest value to the calculated temperature of 1100 K, using the formula \( R = R_0 (1 + \alpha (T - T_0)) \) with the given values of \(\alpha\), \(R_0\), and \(T_0\).

Step-by-step solution

Understand the Formula

The formula for calculating the resistance (R) of a wire at a specific temperature (T) based on the initial resistance (R0) at a reference temperature (T0) is: \( R = R_0 (1 + \alpha (T - T_0)) \) Where: - \(R\) is the resistance at temperature T, - \(R_0\) is the initial resistance at the reference temperature \(T_0\), - \(\alpha\) is the temperature coefficient of resistance, - \(T\) is the temperature we want to find, and - \(T_0\) is the reference temperature. In this exercise, we are given \(\alpha = 0.00125 \, k^{−1}\), \( R_0 = 1 \, \Omega\), and \(T_0 = 300 \, K\). We want to find the temperature \(T\) at which the resistance becomes \(2 \, \Omega\).

Plug in the Given Values

First, plug in the given values into the formula: \(2 = 1 (1 + 0.00125 (T - 300)) \)

Simplify the Equation and Solve for T

Now let's simplify the equation and solve for the temperature, T: \(2 = 1 + 0.00125 (T - 300) \) Subtract 1 from both sides: \(1 = 0.00125 (T - 300)\) Divide both sides by 0.00125: \(\frac{1}{0.00125} = T - 300\) \(800 = T - 300\) Add 300 to both sides: \(800 + 300 = T\) \(T = 1100 \, K\) Now let's check which answer option is closest to 1100 K: (A) 1400 K (B) 1200 K (C) 1000 K (D) 800 K

Choose the Correct Answer

The correct answer is (B) 1200 K, as it is the closest value to the calculated temperature of 1100 K.

Key concepts

Resistance Calculation

Understanding how to calculate resistance at different temperatures is crucial when working with materials like wires. When the resistance of a wire changes, it's often due to a change in temperature. To calculate this change in resistance, we typically use the formula: \[ R = R_0 (1 + \alpha (T - T_0)) \] This equation allows us to determine the resistance \( R \) at a temperature \( T \), given an initial resistance \( R_0 \) at a reference temperature \( T_0 \). The temperature coefficient of resistance, denoted as \( \alpha \), plays a key role here. This variable specifies how much the resistance of a material changes per degree of temperature change. To solve a problem, always start by identifying and substituting the known values into the formula. Then, perform algebraic manipulations to arrive at the unknown variable. This approach is systematic and ensures that calculations are accurate.

Temperature-Resistance Relationship

The relationship between temperature and resistance is a fundamental aspect of electrical conductors. As the temperature of a conductor changes, so does its resistance. This behavior is captured by the temperature coefficient of resistance \( \alpha \), which indicates the rate at which resistance increases with temperature.

• A positive \( \alpha \) means that resistance increases with an increase in temperature, common in most conductors.
• Conversely, a negative \( \alpha \) indicates a decrease in resistance with temperature, though rare in conductors.

This relationship is crucial in selecting materials for electrical components, especially those exposed to temperature variations. It allows engineers and scientists to predict how an electrical circuit will perform under different thermal conditions. By understanding how resistance will change, one can design circuits that function efficiently over a range of temperatures.

Thermal Properties of Conductors

Conductors have unique thermal properties that significantly affect their electrical resistance. Copper, aluminum, and other metals typically used as conductors behave predictably when heated or cooled. As temperature increases, the atoms within the conductor vibrate more energetically, impeding the flow of electrons and increasing resistance. This can be explained by the concept of scattering, where electron collisions become more frequent with increased thermal energy.

• Conductors generally expand when heated; this physical expansion also contributes to an increase in resistance.
• The interaction between resistance and temperature is essential for understanding the overall thermal behavior of a conductor.

By understanding these properties, one can better design and manage devices and systems, ensuring their reliability and efficiency. This knowledge is crucial for applications such as electrical wiring, computer systems, and electronic devices, where consistent performance across a wide range of temperatures is often necessary.