A copper cable is to be designed to carry a current of 280A with a power loss of only 2.70W/m. What is the required radius of the copper cable? Answer in units of cm and use 1.7x10-8 Ωm for the resistivity of copper.
To determine the required radius of the copper cable, we can use the equation for power loss in a wire:
Power Loss = (Resistance) x (Current^2) x (Length)
First, let's rearrange the equation to solve for resistance:
Power Loss = (Resistance) x (Current^2) x (Length)
Resistance = Power Loss / (Current^2 x Length)
Given:
Power Loss = 2.70W/m
Current = 280A
Resistivity of copper (ρ) = 1.7x10^-8 Ωm
Now, we need to calculate the resistance per meter of the copper wire:
Resistance (R) = ρ * (Length / Area)
To calculate the area, we use the formula for the cross-sectional area of a cylinder:
Area = π * (Radius^2)
Now, let's substitute the relevant values into the formulas:
Resistance (R) = ρ * (Length / (π * (Radius^2)))
We can rearrange the equation to solve for the radius:
Radius = sqrt((Resistance * Length) / (π * ρ))
Substituting the given values into the equation:
Radius = sqrt((2.70W/m * 1m) / (π * 280A^2 * 1.7x10^-8 Ωm))
Now let's calculate the value using a calculator:
Radius ≈ 2.52 cm
Therefore, the required radius of the copper cable is approximately 2.52 cm.
To find the required radius of the copper cable, we can use the following formula for the power loss in a conductor:
P = (I^2) * R * L
Where:
P is the power loss (2.70 W/m),
I is the current (280 A),
R is the resistance of the copper cable, and
L is the length of the cable (1 m).
To find the resistance (R), we can use the formula:
R = (ρ * L) / A
Where:
ρ is the resistivity of copper (1.7x10^-8 Ωm),
L is the length of the cable (1 m), and
A is the cross-sectional area of the cable.
We know that the current flows uniformly across the cross-section of the cable, so the current density (J) can be expressed as:
J = I / A
Now, we can rearrange the formula for resistance to solve for the area A:
A = I / J
Using the formula for resistivity, we can express current density (J) as:
J = I / (A * L)
Substituting J into the equation for resistance:
A = I / (ρ * L)
Now, we can substitute the given values into the equation to find the required cross-sectional area (A).
A = 280 A / (1.7x10^-8 Ωm * 1 m)
A ≈ 1.647x10^10 m^2
Finally, we can calculate the radius (r) from the cross-sectional area (A) using the formula:
A = π * r^2
r^2 = A / π
r = √(A / π)
Substituting the value for A:
r ≈ √(1.647x10^10 m^2 / π)
r ≈ 2035.62 m
Converting the radius to centimeters:
r ≈ 2035.62 m * 100 cm/m
r ≈ 203562 cm
Therefore, the required radius of the copper cable is approximately 203562 cm.