**Question:**
We have two copper wires, A and B, with identical mass. The length of wire B is half that of wire A. Find the ratio of the resistances of wire B and wire A. Explain in detail.

**Answer:**

To find the ratio of the resistances of wire B and wire A, we need to understand the factors that affect the resistance of a wire.

**1. Resistance of a wire:**
Resistance is a measure of how much a material opposes the flow of electric current. It depends on several factors, including the material, length, cross-sectional area, and temperature of the wire.

**2. Factors affecting resistance:**
The resistance of a wire can be calculated using Ohm's Law, which states that the resistance (R) is equal to the voltage (V) divided by the current (I), or R = V/I. However, in this case, we are concerned with the factors that affect resistance, not the actual calculation.

- **Material:** Different materials have different resistivities, which is a measure of how strongly a material opposes the flow of electric current. In this case, both wires A and B are made of copper, so they have the same resistivity.
- **Length:** The longer the wire, the higher the resistance. In this case, wire B is half the length of wire A, so its resistance will be half that of wire A.
- **Cross-sectional area:** The larger the cross-sectional area of the wire, the lower the resistance. However, in this case, the cross-sectional area is not given, and we are assuming it to be the same for both wires A and B.
- **Temperature:** The resistance of a wire increases with temperature. However, the temperature is not given in this case, so we can assume it to be constant.

**3. Ratio of resistances:**
Based on the factors mentioned above, we can conclude that the resistance of wire B is half that of wire A because wire B is half the length of wire A. Therefore, the ratio of the resistances of wire B and wire A is 1:2 or 0.5:1.

In summary, the ratio of the resistances of wire B and wire A is 0.5:1 or 1:2. This is because the resistance of a wire is directly proportional to its length, assuming all other factors (material, cross-sectional area, and temperature) remain constant.