Problem 16 - University Physics with Modern 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 will be 9R.

Step 1: Understanding the Relationship of Resistance, Length, and Area

The resistance \( R \) of a wire is given by the formula \( R = \rho \frac{L}{A} \), where \( \rho \) is the resistivity, \( L \) is the length, and \( A \) is the cross-sectional area. To find the new resistance, we must consider how these variables change when the wire is stretched.

Step 2: Express New Length and Cross-sectional Area in Terms of Original Dimensions

When the wire is stretched to three times its original length, the new length \( L' \) becomes \( 3L \), where \( L \) is the original length. Because the volume remains constant, the original volume \( V = L \times A \) must equal the new volume \( V' = 3L \times A' \), where \( A' \) is the new cross-sectional area.

Step 3: Relate Old and New Cross-sectional Areas

Set up the equation for the volumes: \( L \times A = 3L \times A' \). Solving for \( A' \), we find \( A' = \frac{A}{3} \). This indicates that the new cross-sectional area is one-third of the original area.

Step 4: Calculate New Resistance Using New Dimensions

Substitute the new dimensions into the resistance formula: \( R' = \rho \frac{L'}{A'} = \rho \frac{3L}{A/3} = \rho \frac{9L}{A} \). Since the original resistance \( R = \rho \frac{L}{A} \), the new resistance \( R' = 9 \times R \).

Step 5: Final Calculation of New Resistance

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

Key Concepts

Resistivity

Resistivity is a fundamental property of materials that determines how strongly they resist electric current. It is denoted by the symbol \( \rho \) and is measured in ohm-meters (\( \Omega \cdot m \)).

• The resistivity of a material depends on its nature and temperature but remains constant for a given material under specified conditions.
• In mathematical terms, the resistance \( R \) of a uniform wire is related to its resistivity through the formula \( R = \rho \frac{L}{A} \), where \( L \) is the length and \( A \) is the cross-sectional area.

When we stretch a wire, we assume that the resistivity stays the same as long as the material and temperature conditions don't change.
This makes resistivity a key factor in understanding how resistance may vary if the length and area of the wire change.

Ductile Metal Wire

A ductile metal wire is one that can be stretched or drawn into different shapes without breaking. Ductility is a critical property especially when considering changes in wire dimensions such as those discussed in the exercise.

• Ductile metals include copper, aluminum, and silver, all of which are frequently used in electrical wiring.
• The ability to stretch these materials makes them ideal for conducting electricity in various forms and environments.

When a ductile metal wire is stretched to three times its original length, it undergoes a physical transformation, impacting other properties like its cross-sectional area.
The wire's volume stays the same even though its shape changes. This is because the metallic material isn't lost or added, just redistributed along a new, longer length.

Cross-Sectional Area

The cross-sectional area of a wire is the surface area of its slice perpendicular to the length. It significantly impacts the wire's resistance along with its length.

• When a wire is stretched, its length increases and cross-sectional area typically decreases if volume is constant.
• From the exercise, initially, the wire's volume \( V \) is \( L \times A \) and after stretching becomes \( 3L \times A' \).
• Since volume is constant, \( A' \) will be \( \frac{A}{3} \) after stretching, resulting in a smaller area than original.

The cross-sectional area plays an integral part in the formula \( R = \rho \frac{L}{A} \) as it inversely influences resistance.
As the area decreases, the resistance increases, making it an important concept when studying the resistance change in stretched wires.