To determine the ratio of the resistances of the two copper wires with given masses and lengths, we can use the formula for resistance, which is influenced by the material's resistivity, length, and cross-sectional area.
Understanding Resistance
- The resistance (R) of a wire is given by the formula:
R = ρ * (L/A)
Where:
ρ = resistivity of the material
L = length of the wire
A = cross-sectional area
Given Data
- Mass of Wire 1 = 8 gm
- Mass of Wire 2 = 12 gm
- Length Ratio = 3:4
Calculating Volume
- The volume of each wire can be calculated using the formula:
Volume = Mass / Density
- For copper, the density is approximately 8.96 g/cm³.
Cross-Sectional Area Calculation
- Since the volume is proportional to the mass, we can express the volumes in terms of their masses:
Volume 1 = 8/8.96
Volume 2 = 12/8.96
- The cross-sectional area A of each wire can be calculated using:
A = Volume / Length
Finding the Ratio of Resistances
- Let the lengths of the wires be 3x and 4x (based on the given ratio).
- For Wire 1:
A1 = Volume 1 / (3x)
- For Wire 2:
A2 = Volume 2 / (4x)
- Using these relations in the resistance formula, we find:
R1 = ρ * (3x / A1)
R2 = ρ * (4x / A2)
Final Ratio
- The resistances can be expressed in terms of their lengths and cross-sectional areas.
- After simplifying, we find that the ratio of resistances R1:R2 = Length Ratio (3:4) * (Mass Ratio (8:12))
- Thus, the final resistance ratio simplifies to 2:3.
In conclusion, the ratio of the resistances of the two copper wires is 2:3.