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Resistivity is an intrinsic property of a material that measures its opposition to the flow of electric current. It is denoted by the symbol \(\rho\). Imagine resistivity as the inherent friction an electric charge feels as it travels through a wire.
This property is not affected by the size or shape of the material itself. It depends solely on the type of material from which the wire is made. Metals like copper and aluminum have low resistivity, which makes them excellent conductors.

• Resistivity is measured in ohm-meters (\(\Omega \cdot m\)).
• It is affected by temperature, usually increasing as the temperature rises.

In our exercise, when the wire is stretched, the resistivity remains unchanged, which simplifies calculations.

The cross-sectional area of a wire represents the size of its face when cut perpendicular to its length. Think of it as the thickness of the wire. This area plays a crucial role because it determines how much space is available for electrical current to pass through.
In the provided exercise, when the ductile wire is stretched to three times its original length, the cross-sectional area is significantly reduced. This is because the volume of the wire remains constant — stretching increases length while decreasing the cross-sectional area.Understanding this, we find that if the length is tripled, then the new cross-sectional area \(A'\) is reduced to a third of the original (\(A' = \frac{A}{3}\)). Consequently, the smaller the cross-sectional area, the higher the resistance.

A ductile metal wire is one that can be easily stretched or drawn into thin threads without breaking. Metals like copper, aluminum, and gold are examples that exhibit high ductility, making them perfect for wiring and cables.

• Ductility involves the ability to withstand tensile stress.
• It is a property that differs from resistivity, focusing on the material's physical flexibility instead.

In the context of the exercise, the fact that the wire can be stretched to three times its original length means it must be quite ductile. When stretched, the physical dimensions change, but the amount of material remains constant. This affects the geometry but not the material's inherent properties, such as resistivity.

Electrical resistance is a measure of how much a conductor opposes the flow of electric current. The formula for resistance \(R\) is \(R = \rho \cdot \frac{l}{A}\), where \(\rho\) is the resistivity, \(l\) is the length of the conductor, and \(A\) is the cross-sectional area.
To calculate the new resistance when a wire is stretched, we must adjust the values of \(l\) and \(A\) accordingly. In our exercise:

• Original length is \(l\); new length is \(3l\).
• Original area is \(A\); new area is \(\frac{A}{3}\).

Substituting these into the resistance formula gives us the new resistance \(R'\), which is \(R' = \rho \cdot \frac{3l}{\frac{A}{3}} = 9R\). Hence, stretching the wire increases its resistance ninefold.