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 the wire with a diameter of 5.00 mm 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 of the wire,
L is the length of the wire, and
A is the cross-sectional area of the wire.
In this case, we can assume that the resistivity and material of both wires are the same since they are similar. Therefore, we can set up a ratio of resistance using the two wires:
R1/L1 = R2/L2
where:
R1 is the resistance of the first wire,
L1 is the length of the first wire,
R2 is the resistance of the second wire (which we want to find),
L2 is the length of the second wire.
Let's begin by finding the cross-sectional area of the first wire using its diameter:
d1 = 3.00 mm
r1 = d1/2 = 3.00 mm / 2 = 1.50 mm = 0.0015 m
A1 = π * r1^2 = 3.14 * (0.0015 m)^2 = 7.07 x 10^-6 m^2
Now we can substitute these values into the formula for the resistance of the first wire:
R1 = 27.6 Ω
R1 = ρ * (L1 / A1)
L1 = R1 * A1 / ρ
L1 = 27.6 * 7.07x10^-6 / ρ
Since the wires are similar, the ratio of their resistances is equal to the ratio of their lengths. Therefore, we can write:
R1 / L1 = R2 / L2
R2 = R1 * L2 / L1
Substituting the values we know:
R2 = 27.6 * L2 / L1
We are given that L1 = 750 m and R1 = 27.6 Ω. Let's substitute these values into the equation:
R2 = 27.6 * L2 / 750
Now we can solve for the length of the second wire, L2:
L2 = R2 * 750 / 27.6
To find the cross-sectional area of the second wire:
d2 = 5.00 mm
r2 = d2/2 = 5.00 mm / 2 = 2.50 mm = 0.0025 m
A2 = π * r2^2 = 3.14 * (0.0025 m)^2 = 1.96 x 10^-5 m^2
Finally, substituting the values into the equation:
L2 = R2 * 750 / 27.6 = (27.6 * 750) / 27.6 = 750 m
Therefore, the length of the wire with a diameter of 5.00 mm that will have the same resistance is 750 m.