Question

The dependence of drift velocity \( v_d \) on the electric field \( E \), for which Ohm's law is obeyed is

• A. \( v_d \propto E^2 \)
• B. \( v_d \propto E \)
• C. \( v_d \propto \sqrt{E} \)
• D. \( v_d \propto \frac{1}{E} \)
• E. \( v_d \propto \frac{1}{E^2} \)

Hint:

Ohm’s law holds when \( v_d \propto E \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Ohm's law states that the current flowing through a conductor is directly proportional to the voltage across it, provided physical conditions remain constant. At a microscopic level, this law is linked to the movement of charge carriers (electrons) under an applied electric field. The average velocity of these electrons, called drift velocity (\(v_d\)), is key to this relationship.
Step 2: Key Formula or Approach:
The force on an electron in an electric field \(E\) is given by \(F = eE\).
This force causes an acceleration \(a = \frac{F}{m} = \frac{eE}{m}\), where \(m\) is the mass of the electron.
The drift velocity is the average velocity gained by the electron between collisions and is given by \(v_d = a\tau\), where \(\tau\) is the average relaxation time.
Combining these, we get the formula for drift velocity:
\[ v_d = \left(\frac{eE}{m}\right)\tau \] Step 3: Detailed Explanation:
In the expression \(v_d = \frac{e\tau}{m}E\), the charge of the electron (\(e\)), its mass (\(m\)), and the average relaxation time (\(\tau\)) are constants for a given conductor at a constant temperature.
Therefore, the drift velocity \(v_d\) is directly proportional to the electric field \(E\).
\[ v_d \propto E \] This linear relationship is the microscopic foundation of Ohm's Law. Current density \(J\) is defined as \(J = nev_d\). Substituting the expression for \(v_d\) gives \(J = ne\left(\frac{eE\tau}{m}\right) = \left(\frac{ne^2\tau}{m}\right)E\). The term in the bracket is the conductivity \(\sigma\), leading to \(J=\sigma E\), which is Ohm's law in its microscopic form.
Step 4: Final Answer:
For Ohm's law to be obeyed, the drift velocity \(v_d\) must be directly proportional to the electric field \(E\).