If 750 m of 3.00-mm-diameter wire has a resistance of 27.6, what length of similar wire 5.00 mm in diameter will have the same resistance?
To find the length of wire with a different diameter that will have the same resistance, we can use the formula for the resistance of a wire:
R = ρ * (L / A)
Where:
R is the resistance,
ρ (rho) is the resistivity of the wire material,
L is the length of the wire, and
A is the cross-sectional area of the wire.
Given that the resistance of the wire is 27.6 Ω, we can rearrange the formula to solve for L:
L = (R * A) / ρ
First, let's calculate the cross-sectional area (A) of the 3.00-mm-diameter wire. The formula for the area of a circle is:
A = π * r^2
Where:
A is the cross-sectional area,
π (pi) is a mathematical constant approximately equal to 3.14159, and
r is the radius of the wire.
The diameter of the 3.00-mm wire is given, so we can find its radius:
r = 3.00 mm / 2 = 1.50 mm = 0.0015 m
Now, we can calculate the cross-sectional area:
A = π * (0.0015 m)^2
Next, we can use the resistivity of the wire material (ρ) to calculate the length (L) of the second wire.
Finally, we can rearrange the formula to solve for the length of the wire:
L = (R * A) / ρ
Now, let's calculate the length of the second wire using the given values and the formulas explained above.
To find the length of wire with a different diameter that will have the same resistance, we can use the formula for the resistance of a wire:
R = ρ * (L / A)
Where:
R is the resistance,
ρ (rho) is the resistivity of the material the wire is made of,
L is the length of the wire, and
A is the cross-sectional area of the wire.
We are given the values:
R1 = 27.6 Ω (the resistance of the wire),
d1 = 3.00 mm (the diameter of the wire),
and d2 = 5.00 mm (the desired diameter of the wire).
First, let's calculate the cross-sectional area A1 of the wire with a diameter of 3.00 mm:
A1 = π * (d1/2)^2
Next, we'll calculate the length L2 of the wire with a diameter of 5.00 mm:
L2 = (R1 * (d2/2)^2) / (ρ * A1)
Finally, we can substitute the given values into the equation and solve for L2.