91Ó°ÊÓ

Resistivity is a fundamental property of materials that quantifies how strongly a given material opposes the flow of electric current. It is denoted by the symbol \( \rho \) and is expressed in ohm-meters (\( \Omega \cdot m \)).
In simpler terms, resistivity tells us how much resistance a material will provide against electrical conduction.
This property depends on the material's intrinsic nature and remains unchanged unless there is a change in the material itself, such as its temperature.
For example, the resistivity of copper is lower than that of rubber, which is why copper is commonly used in electrical wires.

• Materials with low resistivity are good conductors (e.g., metals).
• Materials with high resistivity are insulators (e.g., rubber).

The length of a wire is one of the key factors that affect its resistance. According to the formula \( R = \rho \frac{L}{A} \), resistance \( R \) is directly proportional to the length \( L \) of the wire.
This means that when the length of the wire increases, so does its resistance, assuming that other factors like resistivity and cross-sectional area remain constant.
In the given exercise, the wire's length was tripled, leading to an increase in resistance.
Thus, we must consider length changes when calculating electrical resistance for wires of different sizes.

• Longer wires have higher resistance because electrons have to travel a greater distance.
• Shorter wires provide less resistance as the path for electron flow is reduced.

The cross-sectional area (A) of a wire is also crucial in determining its resistance. It appears in the denominator of the resistance formula \( R = \rho \frac{L}{A} \), indicating that resistance is inversely proportional to the cross-sectional area.
Simply put, if you increase the area, the resistance decreases, and vice versa. When a wire is stretched and its length increased, its cross-sectional area decreases if the volume is conserved.
For the exercise, as the wire was lengthened to three times its original size, its cross-sectional area decreased to one-third, affecting the overall resistance.

• A larger cross-sectional area allows more current to pass through with less resistance.
• Smaller cross-sectional areas restrict the flow, increasing resistance.

Volume conservation is a principle often applied when dealing with changes in the dimensions of objects, like wires.
The principle states that when an object is deformed, like being stretched, its volume remains constant if the material density is unchanged.
For example, if we extend a wire, making it longer, the cross-sectional area must decrease for the volume to remain constant.
In the exercise, the wire's length increased, which caused the cross-sectional area to decrease, all while retaining its original volume.
This was crucial for calculating the new resistance because it guided us to adjust the area calculation accordingly. Volume conservation helps in maintaining a balance of properties when an object's shape changes.

• If a wire's length triples, its cross-sectional area becomes a third due to constant volume.
• Volume conservation aligns with keeping the material's mass constant.