Drift Velocity

1. Velocity:

The velocity of an electron in a conductor refers to the actual speed and direction of its motion. It is the instantaneous velocity of an electron as it moves randomly in all directions due to thermal energy. This velocity is very high (on the order of ) and does not contribute to the net current.

2. Drift Velocity:

The drift velocity is the average velocity of conduction electrons in a conductor when an electric field is applied. It is the net velocity that contributes to the flow of current in the conductor. The drift velocity is much smaller than the actual velocity of electrons and is typically on the order of .

3. Difference:

The key difference is that while the velocity of an electron is its instantaneous motion in random directions due to thermal energy, the drift velocity is the net average velocity in the direction of the applied electric field, which leads to the flow of current.

We are given a conductor with length and cross-sectional area connected to an ideal battery with emf . The task is to derive an expression for the current density in terms of the relaxation time .

1. **Ohm’s Law:** According to Ohm’s law, the current through a conductor is given by:

Where is the potential difference (emf), and is the resistance of the conductor. The resistance of the conductor is given by:

Where is the resistivity of the material, is the length of the conductor, and is the cross-sectional area of the conductor. Substituting the expression for in Ohm's law:

2. **Current Density:** The current density is defined as the current per unit area of cross-section of the conductor:

Substituting the expression for into this equation:

3. **Relation to Relaxation Time :** The current density can also be expressed in terms of the charge carrier density , the charge of the carriers , the relaxation time , and the drift velocity of the charge carriers:

Where is the number of charge carriers per unit volume, and is the drift velocity of the carriers. The drift velocity is related to the electric field and the relaxation time by:

Where is the mobility of the charge carriers, and is the mass of the charge carriers. The electric field is related to the potential difference and the length of the conductor:

Substituting this into the expression for :

Now, substituting this into the equation for current density :

Thus, the current density in terms of the relaxation time is:

4. **Final Expression for Current Density:** Therefore, the formula for the current density in terms of the relaxation time is:

This equation relates the current density to the emf , the resistivity , and the length of the conductor.

When electrons drift in a conductor from lower to higher potential, it is important to understand the behavior of the free electrons in the conductor. Let's break this down:

1. Electron Drift in a Conductor:

In a conductor, free electrons are constantly moving in random directions due to thermal energy. When an electric field is applied (such as when a potential difference is applied across the conductor), these free electrons experience a force that causes them to "drift" in the direction opposite to the applied electric field. Since electrons are negatively charged, they move from the region of lower potential to the region of higher potential. This is due to the force exerted by the electric field.

2. Are All Free Electrons Moving in the Same Direction?

While the majority of free electrons do indeed drift in the direction from lower to higher potential (opposite to the electric field), it is important to note that not all free electrons are moving in the same direction at any given time.

• Electrons still experience random thermal motion in addition to their drift motion. This random thermal motion means that individual electrons are constantly moving in random directions at high speeds (often much faster than the drift velocity).
• The overall drift of the electrons is a result of their superposition, where the random motion averages out, but the drift due to the electric field is in one direction.
• Thus, while there is a net drift of electrons from lower to higher potential (opposite the electric field), their individual random motions make it so that not all electrons are moving in the same exact direction at every moment.

3. Conclusion:

No, not all the free electrons in a conductor are moving in the same direction when they drift under an applied electric field. While there is a net drift in the direction from lower to higher potential, their random thermal motion causes them to move in various directions. The drift represents a small, net motion superimposed on their random movement.