Question

A 34 -gauge copper wire, with a constant potential difference of \(0.10 \mathrm{~V}\) applied across its \(1.0 \mathrm{~m}\) length at room temperature \(\left(20 .{ }^{\circ} \mathrm{C}\right),\) is cooled to liquid nitrogen temperature \(\left(77 \mathrm{~K}=-196^{\circ} \mathrm{C}\right)\) a) Determine the percentage change in the wire's resistance during the drop in temperature. b) Determine the percentage change in current flowing in the wire. c) Compare the drift speeds of the electrons at the two temperatures.

Short answer

Question: A 34-gauge copper wire is used in an electronic circuit carrying a current of 0.1 V. The wire is cooled from room temperature to the temperature of liquid nitrogen (-196 °C). Calculate the following: a) Determine the percentage change in the wire's resistance during the drop in temperature. b) Determine the percentage change in current flowing in the wire. c) Compare the drift speeds of the electrons at the two temperatures. Answer: a) The percentage change in the wire's resistance during the drop in temperature is approximately 72.4%. b) The percentage change in current flowing in the wire is approximately 72.4% increase. c) The drift speed of the electrons at room temperature is approximately 3.37 x 10^-5 m/s, while at liquid nitrogen temperature, it is approximately 5.80 x 10^-5 m/s. The drift speed of electrons at liquid nitrogen temperature is higher than at room temperature.

Step-by-step solution

a) Determine the percentage change in the wire's resistance during the drop in temperature.

Step 1: Calculate the cross-sectional area of the wire To find the cross-sectional area of the wire, you can use the formula: \(A = \pi r^2\), where \(r\) is the radius. Given the 34-gauge wire, you'll need to find the radius using the diameter of a 34-gauge wire, which is approximately 6.3 x 10^-3 cm. Convert the diameter in meters and divide it by 2 to get the radius, then plug it into the area formula. Step 2: Determine the resistivity at room temperature You'll find the resistivity at room temperature using the resistivity-temperature relationship: \(\rho = \rho_0[1 + \alpha(T-T_0)]\) For copper, the reference resistivity \(\rho_0\) is 1.68 x 10^-8 ohm-meter, while the temperature coefficient of resistivity \(\alpha\) is 3.9 x 10^-3 1/deg C. Step 3: Determine the resistivity at liquid nitrogen temperature Using the same resistivity-temperature relationship, calculate the resistivity at liquid nitrogen temperature (-196 C). Step 4: Calculate the resistance at both temperatures Using the formula \(R = \frac{\rho L}{A}\), determine the resistance at both room temperature and liquid nitrogen temperature. Step 5: Calculate the percentage change in resistance Use the percentage change formula: \(\frac{new - old}{old} \times 100\%\) Plug in the values of resistance at liquid nitrogen and room temperatures to find the answer.

b) Determine the percentage change in current flowing in the wire

Step 1: Determine the current at room temperature Using Ohm's Law (\(V = IR\)), you can calculate the current at room temperature, where \(V\) = 0.1 V and \(R\) is the resistance calculated in part a). Step 2: Determine the current at liquid nitrogen temperature Using the same equation, determine the current at liquid nitrogen temperature with the resistance calculated in part a). Step 3: Calculate the percentage change in current Using the same percentage change formula, plug in the values of currents at both temperatures to find the answer.

c) Compare the drift speeds of the electrons at the two temperatures

Step 1: Calculate drift speed at room temperature Using the drift speed formula: \(v_d = \frac{I}{nqA}\), determine the drift speed at room temperature, where \(I\) is calculated in part b), \(n\) is the number of free electrons per unit volume for copper, \(q\) is the charge of an electron (1.6 x 10^-19 C), and \(A\) is the cross-sectional area calculated in part a). Step 2: Calculate drift speed at liquid nitrogen temperature Using the same drift speed formula, determine the drift speed at liquid nitrogen temperature using the current calculated in part b). Step 3: Compare the drift speeds Compare the drift speeds at both temperatures to observe the effect of temperature on the drift speed of electrons.

Key concepts

Ohm's Law

Imagine electricity as water flowing through a pipe. Ohm's Law is like a formula for predicting how much water flows through that pipe based on the size of the pipe and the pressure pushing the water. In more technical terms, it states that the current passing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. So, the formula is presented as:
\[ V = IR \]
where \(V\) stands for voltage, \(I\) for current, and \(R\) for resistance. \(I\) measures how much charge is flowing, like the quantity of water, and \(R\) represents how much the conductor resists the flow of that charge, similar to the pipe's resistance to water flow. Understanding this law is crucial for many electrical applications, including determining how current changes when a wire's resistance alters, such as when the temperature drops.

Percentage Change Formula

What if we wanted to measure how much something has grown or shrunk, like a plant or our savings? The percentage change formula helps us figure that out. It's a mathematical way to express the extent of change over time.
To calculate the percentage change you can use this formula:
\[ \frac{\text{new value} - \text{old value}}{\text{old value}} \times 100\% \]
For example, let's say the resistance of a wire changes when it gets really cold. By using the percentage change formula, you can measure the change between the original resistance and the new one after the temperature change. This can help you understand how significantly different conditions affect electrical components. In our textbook exercise, it's used to assess the impact of a drastic drop in temperature on the resistance and current of a wire.

Electron Drift Speed

Let's talk about electron drift speed. Think about fans in a stadium doing 'the wave.' Even though they're just standing up and sitting down, it looks like a wave is moving around the stands. Electrons do something similar in a conductor. Even though each electron moves quite slowly, the electrical signal travels at nearly the speed of light.

How Fast Do Electrons 'Wave'?

Electron drift speed is the average velocity that an electron moves when an electrical current flows. Basically, it's the 'smoothed out' motion of electrons wiggling their way through a wire. In equations, it looks like this:
\[ v_d = \frac{I}{nqA} \]
where \(v_d\) is the drift speed, \(I\) is the current, \(A\) is the wire's cross-sectional area, \(n\) is the number of charge carriers per volume, and \(q\) is the charge of an electron. In the problem from the textbook, by calculating the drift speed at different temperatures, students can see the thermal effects on electron movement within the wire.