Question

The drift velocity of electrons in a conducting wire connected to a cell is \( V_d \). If the length of the wire is doubled and area of cross-section is halved, then the drift velocity of electrons becomes:

• A. \( V_d \)
• B. \( \frac{V_d}{2} \)
• C. \( 2V_d \)
• D. \( 4V_d \)

Hint:

When changing dimensions of a wire, remember that resistance depends on length and area: \( R \propto \frac{l}{A} \). Changes in resistance affect the drift velocity accordingly.

Answer 1

The Correct Option is B

The drift velocity \( v_d \) is given by: \[ v_d = \frac{I}{n A e} \] Where: - \( I \) is the current, - \( n \) is the number of charge carriers per unit volume, - \( A \) is the cross-sectional area, - \( e \) is the charge of an electron. When the length \( l \) is doubled and the area \( A \) is halved, the resistance \( R \) of the wire will change. Since \( R \propto \frac{l}{A} \), doubling \( l \) and halving \( A \) will result in a fourfold increase in resistance. Since the current remains the same (as per the conservation of current), the drift velocity \( v_d \) will be halved. Thus, the new drift velocity becomes \( \frac{V_d}{2} \).

Answer 2

Step 1: Understanding drift velocity
Drift velocity \( V_d \) is the average velocity of electrons moving through a conductor when an electric field is applied.

Step 2: Formula for drift velocity
Drift velocity is given by:
\[ V_d = \frac{I}{n e A} \]
where \( I \) is current, \( n \) is number density of electrons, \( e \) is electron charge, and \( A \) is cross-sectional area.

Step 3: Effect of length on resistance and current
Resistance \( R \) of the wire is:
\[ R = \rho \frac{L}{A} \]
where \( \rho \) is resistivity, \( L \) is length, and \( A \) is cross-sectional area.
If length \( L \) is doubled and area \( A \) is halved, resistance becomes:
\[ R' = \rho \frac{2L}{A/2} = 4 \rho \frac{L}{A} = 4R \]

Step 4: Effect on current
With a constant voltage source, current \( I = \frac{V}{R} \), so new current \( I' = \frac{V}{4R} = \frac{I}{4} \).

Step 5: New drift velocity
Since \( V_d \propto \frac{I}{A} \), the new drift velocity is:
\[ V_d' = \frac{I'}{n e A'} = \frac{I/4}{n e (A/2)} = \frac{I}{4} \times \frac{2}{n e A} = \frac{1}{2} \times \frac{I}{n e A} = \frac{V_d}{2} \]

Step 6: Conclusion
Hence, when length is doubled and cross-sectional area is halved, the drift velocity becomes \( \frac{V_d}{2} \).