Problem 13 - College Physics

Question

A ductile metal wire has resistance \(R .\) What will be the resistance of this wire in terms of \(R\) if it is stretched to three times its original length, assuming that the density and resistivity of the material do not change when the wire is stretched. (Hint: The amount of metal does not change, so stretching out the wire will affect its cross-sectional area.)

Step-by-Step Solution

Answer

The resistance becomes \( 9R \).

Step 1: Understand the relationship

The resistance of a wire is given by the formula \( R = \rho \frac{L}{A} \), where \( \rho \) is the resistivity of the material, \( L \) is the length, and \( A \) is the cross-sectional area. Stretching the wire affects \( L \) and \( A \).

Step 2: Relate to original dimensions

Let's assume the original length of the wire is \( L \) and its original cross-sectional area is \( A \). Initially, the resistance is \( R = \rho \frac{L}{A} \).

Step 3: Analyze the new dimensions

When the wire is stretched to three times its original length, its new length \( L' = 3L \). Since the volume of the wire remains constant, \( A' \) will be decreased accordingly.

Step 4: Apply the volume conservation

The volume conservation gives us \( A \times L = A' \times L' \). Substituting \( L' = 3L \), we have \( AL = A' \times 3L \). Therefore, \( A' = \frac{A}{3} \).

Step 5: Substitute into resistance formula

Substitute \( L' \) and \( A' \) into the resistance formula. Thus, \( R' = \rho \frac{3L}{A/3} = \rho \frac{9L}{A} = 9R \).

Step 6: Conclusion

The resistance of the wire after being stretched to three times its original length is \( 9R \).

Key Concepts

Electrical Resistance

Electrical resistance quantifies how strongly a material opposes the flow of electric current. It is a crucial property in electrical systems, determining how easily a current can pass through a component like a wire. The greater the resistance, the harder it is for electricity to flow. The standard unit for measuring resistance is the ohm (\(\Omega\)).

The resistance \(R\) of a wire is influenced by several key factors: resistivity \(\rho\), the length of the wire \(L\), and its cross-sectional area \(A\). These factors combine in the formula:

• \(R = \rho \frac{L}{A}\)

Understanding how each factor affects resistance helps to predict how changes in wire dimensions influence electrical properties. For example, increasing the length \(L\) of a wire, while keeping the cross-sectional area \(A\) constant, results in increased resistance.

Resistivity

Resistivity \(\rho\) is a fundamental property of materials that describes how strongly they resist electric flow. Unlike resistance, which changes with the shape and size of a material, resistivity is an intrinsic property. It is unaffected by these external factors.

Materials with high resistivity require more effort for electrons to pass through compared to materials with low resistivity. Thus, a longer or more resistive material exhibits a higher resistance overall. Resistivity is measured in ohm-meters (\(\Omega\cdot m\)).

When dealing with a wire made of a specific material, knowing its resistivity allows you to calculate expected resistance for different wire configurations, provided you know the length and cross-sectional area.

Cross-sectional Area

The cross-sectional area \( A \) of a wire has a direct impact on its resistance. This area is basically the size of the surface you've cut across the wire, which can be thought of like a slice of bread when you look from the top.

When a wire is stretched, its length increases, but since the total volume of the wire remains the same, the cross-sectional area decreases. In the context of the exercise, if a wire is stretched to three times its original length, the new area \( A' \) becomes one-third of the original area \( A \).

The formula for the resistance, \( R = \rho \frac{L}{A} \), shows that if the cross-sectional area decreases, resistance will increase if all other factors remain constant. This helps in understanding why stretching the wire, as analyzed in the exercise, results in a ninefold increase in resistance.

Length of a Wire

The length \( L \) of a wire is directly proportional to its resistance. As the formula \( R = \rho \frac{L}{A} \) highlights, longer wires have greater resistance because the electrons must travel further, encountering more opposition.

In terms of practical applications, if a wire's length is increased, this additional length requires extra voltage to push the same amount of current through. Moreover, when stretching a wire to elongate its length, while maintaining the same amount of material, the resistance increases significantly.

In the presented exercise, when the wire is stretched to be three times its initial length, the resistance becomes nine times the original due to the decrease in cross-sectional area, demonstrating this concept effectively.