Question

How will the resistivity of a wire change if it is stretched to double its original length without loss of mass?

Answer 1

When a wire is stretched to double its original length without any loss of mass, the resistivity of the material itself does not change. Resistivity is a material property, meaning it is determined by the material of the wire (such as copper, aluminum, etc.), and it doesn't depend on the dimensions (like length or area) of the wire.

However, what changes are the resistance and dimensions of the wire. Here's how:

1. Length (L): When the wire is stretched to double its length, its length increases by a factor of 2. So, if the original length is \( L \), the new length becomes \( 2L \).
   

1. Cross-sectional Area (A): To maintain the same mass (since no mass is lost), the volume of the wire remains constant. The volume is the product of the cross-sectional area \( A \) and the length \( L \). If the length is doubled, the cross-sectional area must be reduced by half to keep the volume constant. So, the new area becomes \( A/2 \).
   

1. Resistance (R): The resistance \( R \) of a wire is given by the formula:
   

   
   \[
   R = \rho \times \frac{L}{A}
   \]

   Since the length \( L \) has doubled and the area \( A \) has halved, the resistance \( R \) will increase by a factor of 4 (because \( R \) is directly proportional to length and inversely proportional to area). This means the resistance will become four times greater.

In summary:

1. The resistivity of the wire remains the same.
   

1. The resistance of the wire increases by a factor of 4.