The electrical resistance of a wire of a given material varies as its length and inversely as the square of the diameter. if a wire 300 feet long and 0.05 inches in diameter has a resistance of 30 ohms, (a) find the resistance of a wire which is 1000 feet long and 0.02 inch in diameter (b) find the length of a wire whose resistance is 20 ohms and diameter is 0.03 inch
R = k L/d^2
given case: L = 300, d = .05, R = 30
30 = k (300/.05^2)
k = .0025(30)/300 = .00025
R = .00025 L/d^2
a) plug in the given values to find R
b) plugging in our values:
20 = .00025 L/.03^2
L = .0009(20)/.00025 = 72
the wire is 72 inches long
To find the resistance of a wire in different scenarios, we need to use the formula that relates the resistance, length, and diameter. Let's denote the resistance as R, the length as L, and the diameter as D.
The formula states that the resistance of a wire varies directly with its length (L) and inversely with the square of its diameter (D). Mathematically, we can express it as:
R ∝ L / D^2
Where ∝ represents proportionality.
Now, let's solve the given problems step by step.
(a) Find the resistance of a wire that is 1000 feet long and 0.02 inches in diameter.
We have the following values:
Length (L1) = 300 feet
Diameter (D1) = 0.05 inches
Resistance (R1) = 30 ohms
We need to find the resistance (R2) of the wire with:
Length (L2) = 1000 feet
Diameter (D2) = 0.02 inches
Using the proportionality equation, we can write:
R1 / R2 = (L1 / L2) * (D2^2 / D1^2)
Substituting the given values, we have:
30 / R2 = (300 / 1000) * (0.02^2 / 0.05^2)
Simplifying this expression gives us:
30 / R2 = 0.3 * 0.0004 / 0.0025
Now solve for R2:
R2 = 30 / (0.3 * 0.0004 / 0.0025)
R2 ≈ 500 ohms
Therefore, the resistance of the wire that is 1000 feet long and 0.02 inches in diameter is approximately 500 ohms.
(b) Find the length of a wire whose resistance is 20 ohms, and diameter is 0.03 inches.
We have the following values:
Resistance (R) = 20 ohms
Diameter (D) = 0.03 inches
Let's assume the length of the wire is L.
Again using the proportionality equation, we can write:
R / 30 = (L / 300) * (0.05^2 / 0.03^2)
Substituting the given values, we have:
20 / 30 = (L / 300) * (0.05^2 / 0.03^2)
Simplifying this expression gives us:
2 / 3 = (L / 300) * (0.0025 / 0.0009)
Now solve for L:
L = 300 * (2 / 3) * (0.0025 / 0.0009)
L ≈ 1852.38 feet
Therefore, the length of the wire with a resistance of 20 ohms and a diameter of 0.03 inches is approximately 1852.38 feet.