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.

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.

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.

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.