Suppose that you wish to fabricate a uniform wire out of 0.900 g of copper. Assume the wire has a resistance R = 0.500 , and all of the copper is used.
(a) What will be the length of the wire? (m)
(b) What will be the diameter of the wire? (µm)
You have two simultaneous equations to solve, and they depend upon the resisitivity and the density, which you will have to look up.
1)0.500 ohms = (resistivity)*L/(pi*D^2/4)
2)0.900 g = (density)*L*pi*(D^2/4)
I suggest using centimeter, g/cm^3 and ohm-cm units.
The product of mass and resistance, which you know, will let you solve for L.
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To find the length of the wire, we can use the formula for resistance:
R = (ρ * L) / A
Where:
R = Resistance (Ω)
ρ = Resistivity of the material (Ω*m)
L = Length of the wire (m)
A = Cross-sectional area of the wire (m²)
First, we need to find the resistivity of copper. The resistivity of copper is approximately 1.7 x 10^-8 Ω*m.
Now, let's calculate the length of the wire:
R = (ρ * L) / A
Solving for L:
L = (R * A) / ρ
To find the cross-sectional area of the wire, we'll need the diameter.
The formula for the cross-sectional area of a wire is:
A = (π * d²) / 4
Where:
A = Cross-sectional area (m²)
π = Pi (approximately 3.14159)
d = Diameter of the wire (m)
To find the diameter of the wire, we'll need to use the density of copper and the mass of the wire.
The density of copper is approximately 8.96 g/cm³.
The formula for the volume of a wire is:
V = (π * d² * L) / 4
Where:
V = Volume (cm³)
π = Pi (approximately 3.14159)
d = Diameter of the wire (cm)
L = Length of the wire (cm)
The formula for density is:
density = mass / volume
Therefore,
volume = mass / density
Now, let's calculate the length of the wire (L):
Given:
mass of copper = 0.900 g
resistance = 0.500 Ω
resistivity of copper = 1.7 x 10^-8 Ω*m
density of copper = 8.96 g/cm³
Step 1: Convert units
Convert the mass of copper from grams to kilograms:
0.900 g = 0.0009 kg
Step 2: Calculate the volume of the wire
volume = mass / density
volume = 0.0009 kg / 8.96 g/cm³
Note: We need to convert cm³ to m³. Since 1 cm = 0.01 m, we have (0.01 m)^3 in the denominator.
Volume = 0.0009 kg / 8.96 g/cm³ * (1 cm / 0.01 m)^3
Volume = 0.0000001 m³
Step 3: Calculate the diameter of the wire
V = (π * d² * L) / 4
We are given the mass of the copper and resistivity, so we can solve for L:
L = (R * A) / ρ
Resolving to A:
A = (R * ρ) / L
Using A in the formula V:
V = (π * d² * (R * ρ) / L) / 4
Now, we can solve for d:
d² = (V * 4 * L) / (π * R * ρ)
d = √((V * 4 * L) / (π * R * ρ))
Substituting the values:
d = √((0.0000001 m³ * 4 * L) / (π * 0.5 Ω * 1.7 x 10^-8 Ω*m))
d = √((0.0000004 L m³) / (π * 0.5 Ω * 1.7 x 10^-8 Ω*m))
d = √(2.35 x 10^-5 L m²)
Step 4: Calculate the length of the wire
L = (R * A) / ρ
Substituting the values:
L = (0.5 Ω * (π * d²) / 4) / (1.7 x 10^-8 Ω*m)
L = ((d² * π * 0.5 Ω) / 4) / (1.7 x 10^-8 Ω*m)
Now, you can solve the equation for L to find the length of the wire.
Note: Due to the complexity of the calculations involved, it would be helpful to use a calculator or a computer software to solve the equations accurately.