Suppose that you wish to fabricate a uniform wire out of 1 g of copper. (Resistivity = 1.72*10^-8 Ohm-m, density = 8.94*10^3kg/m^3). If the wire is to have a resistance of R = 0.5 ohm and all of the copper is to be used, how long is the wire?
a. 1765 m
b. 1816 m
c. 1.8 m
d. 0.5 m
e. 3.3 m
To find the length of the wire, we can use the formula for resistance of a wire:
R = (ρ * L) / A
where R is the resistance, ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
In this case, we know the resistance (R = 0.5 ohm) and the resistivity of copper (ρ = 1.72 * 10^-8 Ohm-m). We also know that the wire is made from 1 g of copper, so we can calculate the volume of the wire using the density of copper (8.94 * 10^3 kg/m^3).
The volume (V) of the wire is given by:
V = mass / density
Since the mass is given as 1 g and the density is given as 8.94 * 10^3 kg/m^3, we can convert the mass to kg:
mass (kg) = 1 g * (1 kg / 1000 g) = 0.001 kg
Now, we can calculate the volume:
V = 0.001 kg / (8.94 * 10^3 kg/m^3) = 1.12 * 10^-7 m^3
We can use the volume to calculate the cross-sectional area of the wire:
A = V / L
Since we want the wire to use all of the copper, we can substitute V with ρ * L * A:
A = (ρ * L * A) / L
Canceling out L, we get:
A = ρ * A
Solving for A, we get:
A = 1.72 * 10^-8 Ohm-m * A
Dividing both sides by 1.72 * 10^-8 Ohm-m, we get:
1 = A
Therefore, the cross-sectional area of the wire is 1 square meter.
Now, we can substitute the values of R, ρ, and A into the resistance formula and solve for L:
0.5 ohm = (1.72 * 10^-8 Ohm-m * L) / 1 square meter
Multiplying both sides by 1 square meter, we get:
0.5 ohm * 1 square meter = 1.72 * 10^-8 Ohm-m * L
Simplifying, we get:
0.5 ohm * 1 square meter = 1.72 * 10^-8 Ohm-m * L
Dividing both sides by 1.72 * 10^-8 Ohm-m, we get:
(0.5 ohm * 1 square meter) / (1.72 * 10^-8 Ohm-m) = L
Calculating this expression, we find:
L = 1.16279 * 10^7 m
Therefore, the length of the wire is approximately 1.16 * 10^7 meters.
None of the given answer options matches this value, so it seems that there might be an error or omission in the question.