Question

What is a wire of resistance R is melted and stretched to N times?

Answer 1

When a wire of resistance \( R \) is melted and then stretched to \( N \) times its original length, its resistance changes. Here’s how to understand it:

1. Understanding Resistance:

The resistance \( R \) of a wire is given by the formula:
\[
R = \rho \frac{L}{A}
\]
Where:

1. \( \rho \) is the resistivity of the material (a constant for a given material),
   

1. \( L \) is the length of the wire,
   

1. \( A \) is the cross-sectional area of the wire.
   

2. Melting and Stretching the Wire:

When the wire is melted and stretched, the total volume of the wire stays constant. The volume before and after stretching remains the same because the material isn't lost—just reshaped.

1. Initially, the volume of the wire is \( V_{\text{initial}} = A_{\text{initial}} \times L_{\text{initial}} \).
   

1. After stretching, the volume remains the same, so:
   

  \[
  A_{\text{final}} \times L_{\text{final}} = A_{\text{initial}} \times L_{\text{initial}}
  \]
  If the length is stretched by a factor of \( N \) (i.e., \( L_{\text{final}} = N \times L_{\text{initial}} \)), the cross-sectional area will decrease by a factor of \( N^2 \) because volume is length × area. Thus, we have:
  \[
  A_{\text{final}} = \frac{A_{\text{initial}}}{N^2}
  \]

3. New Resistance:

Now, using the formula for resistance, the new resistance \( R_{\text{new}} \) after the wire is stretched is:
\[
R_{\text{new}} = \rho \frac{L_{\text{final}}}{A_{\text{final}}}
\]

Substitute the values:
\[
R_{\text{new}} = \rho \frac{N \times L_{\text{initial}}}{\frac{A_{\text{initial}}}{N^2}} = \rho \times \frac{N^3 \times L_{\text{initial}}}{A_{\text{initial}}}
\]

Since the original resistance \( R \) is \( \rho \frac{L_{\text{initial}}}{A_{\text{initial}}} \), we can write:
\[
R_{\text{new}} = N^3 \times R
\]

Final Answer:

The new resistance \( R_{\text{new}} \) is \( N^3 \) times the original resistance \( R \).