Question

The drift velocity of electrons for a conductor connected in an electrical circuit is $V _{ d }$ The conductor in now replaced by another conductor with same material and same length but double the area of cross section The applied voltage remains same The new drift velocity of electrons will be

• A. $\frac{V_{ d }}{4}$
• B. $\frac{ V _{ d }}{2}$
• C. $2 V _{ d }$
• D. $V _{ d }$

Hint:

Drift velocity is independent of the cross-sectional area of the conductor. It depends on the electric field and the relaxation time.

Answer 1

The Correct Option is D

The problem hands us a situation involving the drift velocity of electrons in a conductor. Let's explore how the drift velocity will change under the given conditions.

Initially, the drift velocity for the given conductor is denoted as \(V_d\). The drift velocity in a conductor is related to current, area of cross-section, charge, and electron density by the formula:

\(V_d = \frac{I}{nAe}\)

• \(I\) is the current flowing through the conductor.
• \(n\) is the electron density in the conductor.
• \(A\) is the cross-sectional area of the conductor.
• \(e\) is the charge of an electron.

According to Ohm’s Law, the current \(I\) is:

\(I = \frac{V}{R}\), where \(V\) is the voltage and \(R\) is the resistance.

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

\(R = \frac{\rho L}{A}\)

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

After reorganizing the solution using these two formulas, the drift velocity can be expressed by:

\(V_d = \frac{V}{nAe} \cdot \frac{A}{\rho L} = \frac{V}{n \rho L e}\)

Notice that in this expression, the drift velocity \(V_d\) depends on the applied voltage \(V\), the resistivity \(\rho\), and the length \(L\). Notably, it is independent of the cross-sectional area \(A\) as the area cancels out.

Therefore, even if the area is doubled, the drift velocity of the conductor remains unchanged, provided the material, length, and applied voltage are the same.

Thus, the new drift velocity \(V'_d\) will be:

\(V'_d = V_d\)

Correct Answer: \(V_d\) (The drift velocity remains unchanged)