Introduction:
The resistance of a wire is a measure of its opposition to the flow of electric current. It is influenced by various factors, including the length and radius of the wire. In this explanation, we will explore how the resistance of a wire is affected when both the length and radius are doubled.

Resistance and its formula:
Resistance is determined by the formula R = (ρ * L) / A, where R represents resistance, ρ represents resistivity, L represents length, and A represents cross-sectional area.

Effect of doubling the length:
When the length of a wire is doubled, the resistance also doubles. This can be attributed to the fact that resistance is directly proportional to the length of the wire. As the length increases, the number of collisions between the moving electrons and the atoms in the wire also increases. This results in a higher overall resistance.

Effect of doubling the radius:
When the radius of a wire is doubled, the resistance decreases to one-fourth of its original value. This is because resistance is inversely proportional to the cross-sectional area of the wire. When the radius is doubled, the cross-sectional area becomes four times larger. As a result, there is more space for the electrons to flow, reducing the number of collisions and hence decreasing the resistance.

Combined effect of doubling both length and radius:
If both the length and radius of a wire are doubled simultaneously, their effects on resistance counteract each other.

- Doubling the length increases the resistance by a factor of 2, while
- Doubling the radius decreases the resistance by a factor of 1/4.

Net effect:
To determine the net effect, we can consider the ratios of the new resistance (R') to the original resistance (R):

R' / R = (ρ * 2L) / (4A) = (1/2) * (L / A)

As a result, the net effect of doubling both the length and radius of a wire is that the resistance becomes half of its original value. This means that the resistance decreases by a factor of 2.

Conclusion:
In conclusion, when the length of a wire is doubled, the resistance doubles as well. On the other hand, when the radius is doubled, the resistance decreases to one-fourth of its original value. When both the length and radius are doubled, their effects counteract each other, resulting in the resistance being halved. It is important to consider these factors when designing electrical circuits or analyzing the behavior of wires in various applications.