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. Conductor Resistance:

The resistance \( R \) of a conductor is defined by the equation:

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

Where:

• \( \rho \) denotes the material's resistivity.
• \( l \) represents the conductor's length.
• \( A \) signifies the conductor's cross-sectional area.

Stretching a conductor increases its length and decreases its cross-sectional area, assuming constant volume. If the length is doubled to \( 2l \), the cross-sectional area is reduced to \( A/2 \).

2. Resistance Change Upon Stretching:

Consider initial length \( l \), initial cross-sectional area \( A \), final length \( 2l \), and final cross-sectional area \( A/2 \). The new resistance \( R' \) is calculated as:

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

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

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

Consequently, the final resistance is four times the initial resistance:

\[ R' = 4R \]

3. Drift Velocity:

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

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

Where:

• \( I \) is the current.
• \( n \) is the electron density.
• \( e \) is the electron charge.
• \( A \) is the cross-sectional area.

 

Given constant current \( I \), electron density \( n \), and electron charge \( e \), the drift velocity is inversely proportional to the cross-sectional area:

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

When the conductor is stretched, doubling its length and halving its cross-sectional area, the new drift velocity \( v'_d \) is:

\[ v'_d = 2v_d \]

4. Summary of Findings:

• Resistance relation after stretching: \( R' = 4R \).
• Drift velocity relation after stretching: \( v'_d = 2v_d \).