Question

The relation between the drift velocity (\(V_d\)) and applied electric field (E) of a conductor is

• A. \(V_d \propto E\)
• B. \(V_d \propto \sqrt{E}\)
• C. \(V_d \propto E^2\)
• D. \(V_d \propto \sqrt{E^3}\)

Hint:

Remember that drift velocity is a direct consequence of the force from the electric field. In many simple models, effect is directly proportional to the cause. Here, the electric field \(E\) is the cause, and the drift velocity \(V_d\) is the effect.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field. In a conductor, free electrons move randomly. When an electric field is applied, they experience a force that superimposes a small, directional 'drift' on their random motion.
Step 2: Key Formula or Approach:
The electric field \(E\) exerts a force \(F = eE\) on each free electron (charge \(e\)).
This force causes an acceleration \(a = \frac{F}{m} = \frac{eE}{m}\), where \(m\) is the mass of the electron.
Between collisions with the lattice ions, the electron accelerates. The average time between collisions is called the relaxation time (\(\tau\)).
The drift velocity \(V_d\) is the average velocity gained during this time, given by \(V_d = a\tau\).
Step 3: Detailed Explanation:
Substituting the expression for acceleration into the drift velocity equation: \[ V_d = \left(\frac{eE}{m}\right) \tau \] For a given conductor at a constant temperature, the charge \(e\), mass \(m\), and relaxation time \(\tau\) are all constants.
Therefore, the drift velocity \(V_d\) is directly proportional to the applied electric field \(E\). \[ V_d \propto E \] Step 4: Final Answer:
The drift velocity is directly proportional to the applied electric field. Therefore, option (A) is correct. This relationship is also the microscopic basis for Ohm's law.