Question

What is drift velocity? Derive expression for drift velocity of electrons in a good conductor in terms of relaxation time of electrons?

Answer 1

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Part 1: What is Drift Velocity?

In a metallic conductor (like a copper wire), the outer electrons of the atoms are not bound to individual atoms. They are free to move throughout the entire volume of the metal, forming what is often called an "electron sea" or "electron gas".

1. Motion Without an Electric Field

• In the absence of an external electric field (e.g., when the wire is not connected to a battery), these free electrons are in constant, random motion due to thermal energy.
• They move at very high speeds (around $10^5$ to $10^6$ m/s) and frequently collide with the fixed positive ions (the atomic nuclei) of the metal lattice.
• Because this motion is completely random, an electron is just as likely to move left as it is to move right. Over any period, the net displacement of electrons is zero. Therefore, there is no net flow of charge, and thus no electric current.

2. Motion With an Electric Field

• When an external electric field ($\vec{E}$) is applied across the conductor (e.g., by connecting it to a battery), the situation changes.
• Each free electron experiences an electrostatic force ($\vec{F} = -e\vec{E}$) in the direction opposite to the electric field (since electrons are negatively charged).
• This force accelerates the electrons. However, they don't accelerate indefinitely. Their path is constantly interrupted by collisions with the lattice ions.
• Between collisions, the electron accelerates. At each collision, it loses most of its gained momentum and starts accelerating again in a new random direction, but always with a slight bias from the electric field.

The result is that while the electrons continue their high-speed random motion, the entire "electron sea" is superimposed with a very slow, net "drift" in the direction opposite to the electric field.

Definition: Drift velocity ($v_d$) is the average velocity attained by charged particles (like electrons) in a material due to an electric field.

This velocity is surprisingly slow, typically on the order of millimeters per second ($10^{-4}$ m/s), which is vastly different from the random thermal speeds of the electrons.

Analogy: Imagine a pinball machine that is slightly tilted. The ball (electron) still bounces around frantically and randomly off the pins (ions), but over time, it will gradually make its way to the bottom (drifting in one direction).

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Part 2: Derivation of the Expression for Drift Velocity

Let's derive the expression for the drift velocity ($\vec{v}_d$) of electrons in a good conductor in terms of the relaxation time ($\tau$).

Assumptions and Definitions:

• $\vec{E}$ = The strength of the external electric field applied across the conductor.
• $e$ = The magnitude of the charge on an electron ($1.6 \times 10^{-19}$ C).
• $m$ = The mass of an electron ($9.1 \times 10^{-31}$ kg).
• $\tau$ = Relaxation Time. This is the average time interval between two successive collisions of an electron with the ions of the lattice.

Step-by-Step Derivation:

1. Force on an Electron:
   When an electric field $\vec{E}$ is applied, each free electron experiences a force. According to the Lorentz force law, this force is given by:
   $$ \vec{F} = q\vec{E} $$
   Since the charge of an electron ($q$) is $-e$, the force on an electron is:
   $$ \vec{F} = -e\vec{E} $$
   The negative sign indicates that the force on the electron is in the direction opposite to the electric field.

2. Acceleration of the Electron:
   According to Newton's second law of motion, this force causes the electron to accelerate.
   $$ \vec{F} = m\vec{a} $$
   where $\vec{a}$ is the acceleration of the electron.
   Equating the two expressions for force:
   $$ m\vec{a} = -e\vec{E} $$
   Solving for the acceleration $\vec{a}$:
   $$ \vec{a} = -\frac{e\vec{E}}{m} $$
   This is the constant acceleration an electron experiences between two collisions.

3. Velocity Gained Between Collisions:
   Let's consider the motion of a single electron. Just after a collision, its velocity is random. Let its initial velocity be $\vec{u}_1$. It then accelerates for a time $t_1$ until its next collision. Its final velocity $\vec{v}_1$ just before the next collision can be found using the first equation of motion ($v = u + at$):
   $$ \vec{v}_1 = \vec{u}_1 + \vec{a}t_1 $$
   Similarly, for a second electron:
   $$ \vec{v}_2 = \vec{u}_2 + \vec{a}t_2 $$
   And for the $N^{th}$ electron in the conductor:
   $$ \vec{v}_N = \vec{u}_N + \vec{a}t_N $$

4. Calculating the Average Velocity (Drift Velocity):
   The drift velocity, $\vec{v}_d$, is the average velocity of all ($N$) the free electrons in the conductor.
   $$ \vec{v}_d = \frac{\vec{v}_1 + \vec{v}_2 + \dots + \vec{v}_N}{N} $$
   Substituting the expressions for $\vec{v}_1, \vec{v}_2, \dots$:
   $$ \vec{v}_d = \frac{(\vec{u}_1 + \vec{a}t_1) + (\vec{u}_2 + \vec{a}t_2) + \dots + (\vec{u}_N + \vec{a}t_N)}{N} $$
   We can separate the terms:
   $$ \vec{v}_d = \left(\frac{\vec{u}_1 + \vec{u}_2 + \dots + \vec{u}_N}{N}\right) + \vec{a}\left(\frac{t_1 + t_2 + \dots + t_N}{N}\right) $$

5. Applying the Definitions:
   * The first term, $\frac{\vec{u}_1 + \vec{u}_2 + \dots + \vec{u}_N}{N}$, is the average initial thermal velocity of the electrons. Since their motion is completely random, this average is zero.

   $$ \frac{\vec{u}_1 + \vec{u}_2 + \dots + \vec{u}_N}{N} = 0 $$
   

   • The second term, $\frac{t_1 + t_2 + \dots + t_N}{N}$, is the average time between successive collisions. This is, by definition, the relaxation time, $\tau$.
     $$ \frac{t_1 + t_2 + \dots + t_N}{N} = \tau $$

   Substituting these back into the equation for $\vec{v}_d$:
   $$ \vec{v}_d = 0 + \vec{a}\tau $$
   $$ \vec{v}_d = \vec{a}\tau $$

6. Final Expression:
   Now, substitute the expression for acceleration ($\vec{a} = -e\vec{E}/m$) from Step 2 into this equation:
   $$ \vec{v}_d = \left(-\frac{e\vec{E}}{m}\right)\tau $$

This gives us the final expression for the drift velocity of electrons:

$$ \Large \vec{v}_d = -\frac{e\tau}{m}\vec{E} $$

Key Takeaways from the Expression:

• The drift velocity ($\vec{v}_d$) is directly proportional to the electric field ($\vec{E}$) and the relaxation time ($\tau$).
• The negative sign confirms that the direction of drift for electrons is opposite to the direction of the electric field.
• The magnitude of the drift velocity is given by: $v_d = \frac{e\tau}{m}E$.