Question

The resistance (in ohms) of a circular conductor varies directly with the length of the conductor and inversely with the square of the radius. If 50 feet of wire with a radius of 6 × 10^(-3) inch has a resistance of 10 ohms, what would be the resistance of 100 feet of the same wire if the radius were increased to 7 × 10^(-3) inch?

Answer 1

The resistance of a circular conductor varies directly with the length and inversely with the square of the radius. By applying the given value of resistance and radius for a 50-foot wire, the constant of proportionality can be determined. Using this constant, the resistance of a 100-foot wire with a different radius can be calculated.

Step-by-step explanation:

The resistance (R) of a circular conductor is directly proportional to the length (L) of the conductor and inversely proportional to the square of the radius (r). Mathematically, it can be expressed as: R = k(L/r^2), where k is the constant of proportionality.

Given that 50 feet of wire with a radius of 6 × 10^(-3) inch has a resistance of 10 ohms, we can substitute these values into the formula:

1. Convert 50 feet to inches: 50 feet * 12 inches/foot = 600 inches.
2. Convert the radius to inches: 6 × 10^(-3) inch.
3. Substitute these values into the formula: 10 ohms = k(600 inches/(6 × 10^(-3) inch)^2).
4. Solve for k: k = 10 ohms * (6 × 10^(-3) inch)^2 / 600 inches = 0.06.
5. Now we can use k to find the resistance for 100 feet of wire with a radius of 7 × 10^(-3) inch.
6. Convert 100 feet to inches: 100 feet * 12 inches/foot = 1200 inches.
7. Convert the new radius to inches: 7 × 10^(-3) inch.
8. Substitute these values into the formula: R = 0.06 * (1200 inches/(7 × 10^(-3) inch)^2).
9. Simplify and calculate: R = 0.06 * (1200 inches/49 × 10^(-6) inch^2) = 2.45 ohms.

Therefore, the resistance of 100 feet of wire with a radius of 7 × 10^(-3) inch would be approximately 2.45 ohms.