Copper wire length calculation to achieve a resistance of 10 ohm:

Given data:
Cross-sectional area of the wire, A = 4×10^-6 m^2
Resistivity of copper, ρ = 1.6×10^-8 ohm m
Desired resistance, R = 10 ohm

To calculate the length of the wire required to achieve a resistance of 10 ohm, we can use the formula:

R = ρ * (L / A)

Where:
R is the resistance,
ρ is the resistivity,
L is the length of the wire, and
A is the cross-sectional area of the wire.

Rearranging the formula, we get:

L = (R * A) / ρ

Substituting the given values, we have:

L = (10 * 4×10^-6) / (1.6×10^-8)

L = (40×10^-6) / (1.6×10^-8)

L = 25 meters

Therefore, to achieve a resistance of 10 ohm, the length of the copper wire should be 25 meters.

Effect of doubling the diameter on resistance:

The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area. When the diameter (or radius) of a wire is doubled, the cross-sectional area is increased by a factor of 4 (since area is proportional to the square of the radius).

Let's assume the original diameter of the wire is d, and its original resistance is R1. When the diameter is doubled, the new diameter becomes 2d, and the new resistance becomes R2.

Using the formula for resistance, we know that:

R = ρ * (L / A)

Since the resistivity (ρ) and length (L) of the wire remain constant, we can compare the changes in resistance by comparing the changes in cross-sectional area.

The original resistance, R1 = ρ * (L / A1)

The new resistance, R2 = ρ * (L / A2)

Since A2 is 4 times larger than A1 (due to doubling the diameter), we can write A2 = 4A1.

Substituting this into the equation for R2, we get:

R2 = ρ * (L / 4A1)

R2 = (1/4) * ρ * (L / A1)

R2 = (1/4) * R1

Therefore, when the diameter is doubled, the resistance becomes one-fourth of its original value. In other words, the resistance decreases by 75%.

Explanation:

- The length of the wire required to achieve a resistance of 10 ohm is calculated using the formula R = ρ * (L / A).
- Substituting the given values, we find that the length should be 25 meters.
- Doubling the diameter of the wire increases the cross-sectional area by a factor of 4, resulting in a decrease in resistance by 75%.
- This relationship can be understood by considering the formula R = ρ * (L / A), where resistance is inversely proportional to cross-sectional area.
- The resistivity (ρ) and length (L) of the wire remain constant in this scenario.
- Therefore, doubling the diameter of the wire leads to a significant decrease in resistance