Question

The resistance of a resistance thermometer has values \(2.71\) and \(3.70 \mathrm{ohm}\) at \(10^{\circ} \mathrm{C}\) and \(100^{\circ} \mathrm{C}\). The temperature at which the resistance is \(3.26 \mathrm{ohm}\) is (A) \(40^{\circ} \mathrm{C}\) (B) \(50^{\circ} \mathrm{C}\) (C) \(60^{\circ} \mathrm{C}\) (D) \(70^{\circ} \mathrm{C}\)

Short answer

The temperature at which the resistance is \(3.26 \mathrm{ohm}\) is \(60^{\circ}\mathrm{C}\).

Step-by-step solution

Finding the slope of the linear relation between temperature and resistance

First, we will use the given data points to find the slope between temperature and resistance. The slope can be calculated using the formula: Slope = \(\frac{Change\ in\ Resistance}{Change\ in\ Temperature}\) So, Slope = \(\frac{3.70 \mathrm{ohm} - 2.71 \mathrm{ohm}}{100^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C}}\)

Calculate the slope value

Now, let's calculate the slope using the given values: Slope = \(\frac{3.70 \mathrm{ohm} - 2.71 \mathrm{ohm}}{100^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C}}\) = \(\frac{0.99 \mathrm{ohm}}{90^{\circ} \mathrm{C}}\) = \(0.011 \mathrm{ohm}/^{\circ}\mathrm{C}\)

Finding the linear equation relating temperature and resistance

With the slope value, we can now find the linear equation that relates temperature (T) and resistance (R). Using one of the given data points (let's use \(10^{\circ}\mathrm{C}\) and \(2.71 \mathrm{ohm}\)): \(R - R_1 = Slope \times (T - T_1)\) where \(R_1\) is the known resistance value, and \(T_1\) is the known temperature value. Now, let's plug in the values. We have: \(R - 2.71 \mathrm{ohm} = 0.011 \frac{\mathrm{ohm}}{^{\circ}\mathrm{C}} \times (T - 10^{\circ}\mathrm{C})\)

Finding the temperature corresponding to the given resistance

We are given the resistance value as \(3.26\mathrm{ohm}\). Now, we will plug this into our equation and solve for the temperature (T): \(3.26 \mathrm{ohm} - 2.71 \mathrm{ohm} = 0.011\frac{\mathrm{ohm}}{^{\circ}\mathrm{C}} \times (T - 10^{\circ}\mathrm{C})\) Next, we will simplify and solve for T: \(0.55 \mathrm{ohm} = 0.011\frac{\mathrm{ohm}}{^{\circ}\mathrm{C}} \times (T - 10^{\circ}\mathrm{C})\) To solve for T, we will divide both sides by the slope: \(\frac{0.55\mathrm{ohm}}{0.011\frac{\mathrm{ohm}}{^{\circ}\mathrm{C}}} = T - 10^{\circ}\mathrm{C}\) Now, we can calculate the value of T: \(T = 10^{\circ}\mathrm{C} + \frac{0.55\mathrm{ohm}}{0.011\frac{\mathrm{ohm}}{^{\circ}\mathrm{C}}} = 10^{\circ}\mathrm{C} + 50^{\circ}\mathrm{C} = 60^{\circ}\mathrm{C}\) So, the temperature at which the resistance is \(3.26 \mathrm{ohm}\) is \(60^{\circ}\mathrm{C}\), which corresponds to option (C).

Key concepts

Understanding the Linear Relationship Between Resistance and Temperature

A resistance thermometer works by measuring the resistance of a material, often metal, which changes with temperature. This change occurs in a linear fashion, meaning that as the temperature increases or decreases, the resistance adjusts in a straight-line correlation. The linear relationship can be expressed mathematically as: - Resistance (R) = Slope × Temperature (T) + Intercept. This essentially means that for every degree of temperature change, there's a corresponding predictable change in resistance. By observing how the resistance of a thermometer changes with known temperatures, such as those provided in an exercise - e.g., at 10°C and 100°C - you can identify the characteristics of this linear relationship using the slope, which helps predict resistance at other temperatures.

How to Calculate the Slope in Resistance vs. Temperature

To find the slope, you use two known data points of temperature and resistance. The formula for slope is: \[\text{Slope} = \frac{\text{Change in Resistance}}{\text{Change in Temperature}}\] In the exercise, we calculated this using the points (10°C, 2.71 ohm) and (100°C, 3.70 ohm). The change in resistance is 3.70 - 2.71, and the change in temperature is 100 - 10. Thus, the slope turns out to be 0.011 ohm/°C. This slope helps us understand how much the resistance changes with a one-degree shift in temperature. It is a crucial step in establishing the linear relationship, as it defines the rate of change of resistance per unit temperature.

Deriving the Resistance-Temperature Equation

With the slope known, you can write the linear equation relating temperature (T) to resistance (R). The general formula here becomes: \[R - R_1 = \text{Slope} \times (T - T_1)\] where \(R_1\) and \(T_1\) are known resistance and temperature values. For the given problem, we used the point (10°C, 2.71 ohm) to derive the equation: - \(R - 2.71 = 0.011 \times (T - 10)\) This equation enables us to calculate resistance for any given temperature or vice versa. Solving this equation for other resistances helps find unknown temperatures, like solving it for a resistance of 3.26 ohm in the exercise, which roughly corresponds to 60°C.

Applying Problem Solving in Physics

Solving physics problems, especially those involving linear equations, requires a strategic approach. Follow these steps with accuracy: - Identify known values: Start with what you have been given, like specific resistance and temperature data points. - Calculate slope: Use the data points to find the rate of change or slope, which describes how one quantity changes with the other. - Derive an equation: Use the slope and known values to form an equation relating the variables of interest. - Solve for unknowns: Substitute any given value in your equation to find the unknown variable of interest. This structured method not only simplifies complex problems but also enhances your problem-solving skills in physics. Remember to ensure calculations are precise, and each step logically follows the previous.