Question

A conductor of length \( l \) is connected across an ideal cell of emf \( E \). Keeping the cell connected, the length of the conductor is increased to \( 2l \) by gradually stretching it. If \( R \) and \( R' \) are the initial and final values of resistance, and \( v_d \) and \( v'_d \) are the initial and final values of drift velocity, find the relation between (i) \( R' \) and \( R \) and (ii) \( v'_d \) and \( v_d \).

Hint:

Stretching a conductor doubles its length and halves its cross-sectional area, leading to a fourfold increase in resistance and a doubling of the drift velocity to maintain the current.

Answer 1

1. Resistance of a Conductor: 

The resistance \( R \) of a conductor is given by the formula:

\[ R = \rho \frac{l}{A} \]

Where:

• \( \rho \) is the resistivity of the material.
• \( l \) is the length of the conductor.
• \( A \) is the cross-sectional area of the conductor.

 

When the conductor is stretched, its length increases, and the cross-sectional area decreases (since the volume of the conductor remains constant during stretching). If the length of the conductor is doubled to \( 2l \), the cross-sectional area \( A \) becomes half of its original value.

2. Relation Between Initial and Final Resistance (\( R' \) and \( R \)):

Let the initial length be \( l \) and the initial cross-sectional area be \( A \), and let the final length be \( 2l \) and the final cross-sectional area be \( A/2 \). The resistance after stretching the conductor is given by:

\[ R' = \rho \frac{2l}{A/2} = \rho \frac{4l}{A} \]

Thus, the ratio of the final resistance \( R' \) to the initial resistance \( R \) is:

\[ \frac{R'}{R} = \frac{\frac{4l}{A}}{\frac{l}{A}} = 4 \]

Therefore, the relation between the final and initial resistance is:

\[ R' = 4R \]

3. Drift Velocity:

The drift velocity \( v_d \) of the electrons is given by the equation:

\[ v_d = \frac{I}{n A e} \]

Where:

• \( I \) is the current flowing through the conductor.
• \( n \) is the number of free electrons per unit volume of the material.
• \( e \) is the charge of an electron.
• \( A \) is the cross-sectional area of the conductor.

 

Since the current \( I \) is constant (because the emf \( E \) and resistance \( R \) are constant), and \( n \) and \( e \) are constant, we observe that the drift velocity is inversely proportional to the cross-sectional area of the conductor:

\[ v_d \propto \frac{1}{A} \]

When the length of the conductor is doubled, the cross-sectional area is halved. Therefore, the drift velocity after stretching becomes:

\[ v'_d = 2v_d \]

4. Conclusion:

• Relation between final and initial resistance: \( R' = 4R \).
• Relation between final and initial drift velocity: \( v'_d = 2v_d \).