A 50.0-g sample of a conducting material is all that is available. The
resistivity of the material is measured to be 11 x 10-8 ohms/m and the density is 7.86 g/cm3. The
material is to be shaped into a solid cylindrical wire that has a total resistance of 1.5 ohms.
a) What length of wire is required?
b) What must be the diameter of the wire?
m=0.05 kg, ρ =11•10⁻⁸ Ω/m
d=7860 kg/m³ R=1.5Ω
R= ρ•L/A
d=m/V=m/A•L
R/d= ρ•L²•A/A•m = ρ•L²/m.
L=sqrt{m•R/ρ•}=…
A= ρ•L/R = …
A=πD²/4
D=sqrt(4A/π)=…
a) To find the length of the wire required, we can use the resistivity formula:
Resistivity (ρ) = (Resistance x Area) / Length
Given that the resistance is 1.5 ohms and the resistivity is 11 x 10^-8 ohms/m, we can rearrange the formula:
Length = (Resistance x Area) / Resistivity
The area of a cylinder is given by:
Area = π x (Radius)^2
First, let's convert the density from g/cm³ to kg/m³:
Density = 7.86 g/cm³ = 7860 kg/m³
To find the volume of the wire, we can use the formula:
Volume = Mass / Density
Mass = 50.0 g
Volume = 50.0 g / 7860 kg/m³ = 0.00636 m³
Next, we need to calculate the radius of the wire. Since we don't have the diameter, we'll use the formula:
Volume = π x (Radius)^2 x Length
Length = Volume / (π x (Radius)^2)
Since we don't know the radius or length, we can't directly calculate one from the other. However, we can use the given resistance to find a relationship between them.
Resistance = Resistivity x (Length / Area)
1.5 ohms = 11 x 10^-8 ohms/m x (Length / (π x (Radius)^2))
Let's call the radius R and the length L for now.
1.5 = 11 x 10^-8 x (L / (π x R^2))
Rearranging the equation:
L = (1.5 x π x R^2) / (11 x 10^-8)
Now, let's substitute the volume equation for R^2:
L = (1.5 x π x Volume) / (11 x 10^-8 x Area)
L = (1.5 x π x 0.00636 m³) / (11 x 10^-8 x π x (Radius)^2)
L = (1.5 x 0.00636) / (11 x 10^-8)
L = 0.00954 / (11 x 10^-8)
L ≈ 0.00954 x 10^8 / 11
L ≈ 8.67 x 10^5 m
b) We have the length of the wire, and we can use it to find the radius (diameter) of the wire.
Length = π x (Radius)^2
0.00954 x 10^8 m = π x (Radius)^2
Rearranging the equation:
Radius^2 = (0.00954 x 10^8) / π
Radius^2 = 3.043 x 10^7
Radius ≈ √(3.043 x 10^7)
Radius ≈ 5516.4 m
Diameter ≈ 2 x Radius ≈ 11032.8 m
So, the approximate diameter of the wire should be around 11032.8 meters.
Please note that these calculations assume the wire is perfectly cylindrical, and it may not be realistic or practical in real life. I hope you found this amusing!
To find the length of wire required and the diameter of the wire, we can use the following formulas:
a) The resistance of the wire is given by the formula: R = (ρ * L) / A
b) The cross-sectional area of the wire is given by the formula: A = π * (d/2)^2
Now let's solve these step-by-step:
Step 1: Convert the resistivity from ohms/m to ohms/cm.
- Given resistivity = 11 x 10^-8 ohms/m
- Convert to ohms/cm: 11 x 10^-8 ohms/m * (1 m / 100 cm) = 1.1 x 10^-9 ohms/cm
Step 2: Calculate the mass of the wire.
- Given mass = 50.0 g
Step 3: Calculate the volume of the wire.
- The density is given as 7.86 g/cm^3, and the mass is given in grams.
- Volume = mass / density = 50.0 g / 7.86 g/cm^3
Step 4: Convert the volume from cm^3 to m^3.
- 1 cm^3 = (1/100)^3 m^3 = 10^-6 m^3.
- Volume = (50.0 g / 7.86 g/cm^3) * (10^-6 m^3 / 1 cm^3)
Step 5: Calculate the length of the wire.
- Given total resistance = 1.5 ohms.
- Given resistivity = 1.1 x 10^-9 ohms/cm.
- Using the formula R = (ρ * L) / A, where A = π * (d/2)^2, we substitute the values:
1.5 ohms = (1.1 x 10^-9 ohms/cm * L) / (π * (d/2)^2)
- Rearranging the equation, we find: L = (1.5 ohms * π * (d/2)^2) / (1.1 x 10^-9 ohms/cm)
Step 6: Solve for the length.
- Plug in the values and calculate the length: L = (1.5 ohms * π * (d/2)^2) / (1.1 x 10^-9 ohms/cm)
Step 7: Calculate the diameter of the wire.
- Using the formula A = π * (d/2)^2, rearrange the equation to solve for d:
d = √((4A)/π)
- Plug in the value of A from step 4 and calculate: d = √((4A)/π)
By following these steps, you should be able to calculate the length of wire required and the diameter of the wire.
To answer these questions, we need to use the formulas for resistivity, resistance, and the geometry of a cylindrical wire.
a) First, let's calculate the volume of the wire using the given density. The volume of a cylindrical wire can be calculated using the formula:
Volume = (pi * r^2 * h),
where r is the radius and h is the height or length of the wire.
Since the density is given in grams per cubic centimeter (g/cm^3), we need to convert it to grams per cubic meter (g/m^3) by dividing by 1000.
Density = 7.86 g/cm^3 = 7.86 * (1000 g/m^3) = 7860 g/m^3.
The mass of the wire is given as 50.0 grams.
Mass = 50.0 g.
The volume can be calculated using the formula:
Volume = Mass / Density,
Volume = 50.0 g / 7860 g/m^3 = 0.0064 m^3.
Next, we need to calculate the length of the wire. We can rearrange the formula for the volume of a cylinder:
Volume = (pi * r^2 * h) => h = Volume / (pi * r^2).
The resistance of the wire is given as 1.5 ohms.
Resistance = 1.5 ohms.
The resistivity of the material is also given as 11 x 10^-8 ohms/m.
Resistivity = 11 x 10^-8 ohms/m.
The resistance of a wire can be calculated using the formula:
Resistance = (Resistivity * Length) / Area,
where Length is the length of the wire, and Area is the cross-sectional area of the wire.
By rearranging the formula, we can solve for Length:
Length = (Resistance * Area) / Resistivity.
From here, we can substitute the known values:
Length = (1.5 ohms * Area) / (11 x 10^-8 ohms/m).
b) To find the diameter of the wire, we can use the formula for the area of a circle:
Area = pi * r^2.
From the formula for Length, we can solve for Area:
Area = (Length * Resistivity) / Resistance.
Then, we can rearrange the formula for Area to solve for the radius:
r = sqrt(Area / pi).
From the radius, we can calculate the diameter using the formula:
Diameter = 2 * radius.
Let's calculate the values step by step:
- Calculate the volume of the wire:
Volume = Mass / Density = 50.0 g / 7860 g/m^3 = 0.0064 m^3.
- Calculate the length of the wire using the formula:
Length = (1.5 ohms * Area) / (11 x 10^-8 ohms/m).
- Calculate the area of the wire using the formula:
Area = (Length * Resistivity) / Resistance.
- Calculate the radius of the wire using the formula:
r = sqrt(Area / pi).
- Calculate the diameter of the wire:
Diameter = 2 * radius.