A 51.0 g sample of a conducting material is all that is available. The resistivity of the material is measured to be 12 10-8 · m, and the density is 7.67 g/cm3. The material is to be shaped into a solid cylindrical wire that has a total resistance of 1.5 .
What length of wire is required?
What must be the diameter of the wire?
Is the 12*10^-8 resistivity in units of ohm-cm and the wire resistance in ohms? You have omitted some symbols.
The 12*10^-8 is in ohm-m, and the wire resistance is in ohms. I copy pasted the question from "WebAssign" so I guess the symbol for ohms did not transfer over.
To determine the length of wire required, we can use the formula for resistance:
Resistance (R) = Resistivity (ρ) * Length (L) / Cross-sectional Area (A)
Given:
Resistance (R) = 1.5 Ω
Resistivity (ρ) = 12 * 10^-8 Ω·m
To find the length (L), we need to determine the cross-sectional area (A) of the wire.
First, let's calculate the mass of the wire using the density:
Density = Mass / Volume
Volume = Mass / Density
Given:
Density = 7.67 g/cm³
Mass = 51.0 g
Converting the density to kg/m³ and the mass to kg:
Density = 7.67 * 1000 kg/m³
Mass = 0.051 kg
Now we can find the volume:
Volume = Mass / Density
Substituting the values:
Volume = 0.051 kg / (7.67 * 1000 kg/m³) = 6.647 * 10^(-6) m³
The cross-sectional area (A) of a cylinder can be calculated using the formula:
Area (A) = π * (diameter/2)²
To find the diameter, we need to rearrange the formula for resistance:
R = ρ * L / A
Solving for A:
A = ρ * L / R
Substituting the known values:
A = (12 * 10^-8 Ω·m) * L / (1.5 Ω)
Now we can substitute the value of A into the formula for the cross-sectional area of the cylinder:
π * (diameter/2)² = (12 * 10^-8 Ω·m) * L / (1.5 Ω)
Simplifying:
π * (diameter/2)² = 8 * 10^-8 Ω·m * L
π * (diameter/2)² = 8 * 10^-8 Ω·m * L
Now, to find the diameter, we need to rearrange the formula:
(diameter/2)² = (8 * 10^-8 Ω·m * L) / π
Substituting the values and solving for (diameter/2)²:
(diameter/2)² = (8 * 10^-8 Ω·m * L) / π
Finally, we can solve for the length (L) using the desired resistance:
1.5 Ω = (12 * 10^-8 Ω·m) * L / A
Substituting the calculated value of A, we can solve for L:
1.5 Ω = (12 * 10^-8 Ω·m) * L / [(8 * 10^-8 Ω·m * L) / π]
Simplifying:
1.5 Ω = π * L² / 8
L² = (1.5 Ω * 8) / π
Solving for L:
L = √[(1.5 Ω * 8) / π]
Please note that this calculation assumes a uniform material and ideal conditions.