Question

When a wire of resistance r is cut into two?

Answer 1

When a wire with resistance \( r \) is cut into two parts, the resistance of each part depends on how the wire is divided.

Key points to remember:

1. Resistivity: The resistance of a wire is given by the formula:
   

   \[
   R = \rho \cdot \frac{L}{A}
   \]
   where:
   - \( R \) is the resistance,
   - \( \rho \) is the resistivity of the material,
   - \( L \) is the length of the wire,
   - \( A \) is the cross-sectional area of the wire.

1. If the wire is cut in half (i.e., into two equal parts):
   

   - The length of each part will be half of the original length.
   - Since resistance is directly proportional to length, the resistance of each part will be half of the original resistance \( r \).

   Example:  
   If the wire of resistance \( r \) is cut into two equal pieces, each piece will have a resistance of \( \frac{r}{2} \).

1. If the wire is cut in unequal lengths:
   

   - The resistance of each part will be proportional to its length. If one part is longer than the other, its resistance will be greater.

   If the wire is cut into two parts where one piece is of length \( L_1 \) and the other is \( L_2 \), the resistance of each part will be:
   \[
   R_1 = r \cdot \frac{L_1}{L_1 + L_2} \quad \text{and} \quad R_2 = r \cdot \frac{L_2}{L_1 + L_2}
   \]
   where \( r \) is the total original resistance, and \( L_1 + L_2 \) is the original length of the wire.

To sum up:

1. Equal parts: Each part's resistance = \( \frac{r}{2} \).
   

1. Unequal parts: Each part's resistance depends on the ratio of the lengths.