Resistance of a Wire

The resistance of a wire is a measure of how much it opposes the flow of electric current. It is determined by the material of the wire, its length, and its cross-sectional area. The resistance is directly proportional to the length of the wire and inversely proportional to its cross-sectional area. Mathematically, it can be expressed as:

R = ρ * (L/A)

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

Effect of Radius on Resistance

According to the equation above, the resistance of a wire is inversely proportional to its cross-sectional area (A). This means that as the radius of the wire decreases, the resistance increases. Conversely, if the radius of the wire increases, the resistance decreases.

Halving the Radius

When the radius of a wire is halved, the cross-sectional area is reduced to one-fourth of its original value. This is because the area of a circle is proportional to the square of its radius. Therefore, if the radius is halved, the area becomes one-fourth.

Calculating the New Resistance

To calculate the new resistance (R'), we can substitute the new values into the resistance equation:

R' = ρ * (L/A')

Where:
R' is the new resistance of the wire
ρ is the resistivity of the material
L is the length of the wire
A' is the new cross-sectional area of the wire

Since the radius is halved, the new cross-sectional area can be calculated as:

A' = (π * r'^2)

Where:
r' is the new radius of the wire

Substituting this value into the equation, we get:

R' = ρ * (L / (π * r'^2))

Conclusion

In conclusion, when the radius of a wire is halved, its resistance increases. This is because the resistance of a wire is inversely proportional to its cross-sectional area. Halving the radius reduces the cross-sectional area to one-fourth of its original value, resulting in a higher resistance.