Problem Analysis

We are tasked with finding the resistance of a second wire given the following conditions:

1. The first wire has a resistance $R_1 = 2 \, \Omega$, a length $L_1 = 1 \, \text{m}$, and a specific resistivity $\rho$.
2. The second wire has:
   • Length: $L_2 = 3 \, \text{m}$,
   • Diameter: Twice that of the first wire ($d_2 = 2d_1$),
   • Specific resistance (resistivity): $\rho_2 = 2\rho_1$ (double the resistivity of the first wire).

The resistance $R$ of a wire is determined using the formula: $R = \rho \frac{L}{A}$ where:

• $\rho$ is the resistivity,
• $L$ is the length,
• $A$ is the cross-sectional area ($A = \pi r^2$, where $r$ is the radius).

Step-by-Step Solution

1. First Wire's Characteristics: For the first wire: $R_1 = \rho_1 \frac{L_1}{A_1}$ Substituting $L_1 = 1 \, \text{m}$, $A_1 = \pi r_1^2$: $R_1 = \rho_1 \frac{1}{\pi r_1^2}$ Given $R_1 = 2 \, \Omega$: $\rho_1 = 2 \pi r_1^2$

2. Second Wire's Characteristics: For the second wire:

   • Resistivity $\rho_2 = 2\rho_1$,
   • Length $L_2 = 3 \, \text{m}$,
   • Radius $r_2 = 2r_1$, so cross-sectional area $A_2 = \pi r_2^2 = \pi (2r_1)^2 = 4\pi r_1^2$.

   Substituting these into the resistance formula: $R_2 = \rho_2 \frac{L_2}{A_2}$ Substitute $\rho_2 = 2\rho_1$, $L_2 = 3$, $A_2 = 4\pi r_1^2$: $R_2 = (2\rho_1) \frac{3}{4\pi r_1^2}$ Substitute $\rho_1 = 2 \pi r_1^2$ from the first wire: $R_2 = 2(2\pi r_1^2) \frac{3}{4\pi r_1^2}$ Simplify: $R_2 = 3 \, \Omega$

Final Answer:

The resistance of the second wire is: $\boxed{3 \, \Omega}$

Would you like a more detailed explanation of any specific step or assistance with related problems? Here are 5 related questions to explore:

1. How does the resistivity affect the resistance of a wire?
2. What happens to resistance if only the length of the wire changes?
3. How is resistance affected if both diameter and resistivity change inversely?
4. What role does temperature play in the resistivity of a material?
5. Can the resistance formula be experimentally verified? How?

Tip: Always consider the dependency of resistance on both geometric and material properties while solving wire-related problems.