Relation between drift velocity of electron, thermal velocity is given by :

Options

A. `v_(r) = v_(d) = 0`
B. `v_(d) = v_(r)`
C. `v_(d) gt v_(r)`
D. `v_(d) lt v_(r)`

Explanation

To establish the relationship between the drift velocity of electrons and their thermal velocity, we can follow these steps:

### Step 1: Understand Drift Velocity
Drift velocity (\(v_d\)) is the average velocity that a charge carrier, such as an electron, attains due to an electric field. It is defined as the net velocity of the electrons when an external voltage is applied across a conductor.

### Step 2: Understand Thermal Velocity
Thermal velocity (\(v_t\)) refers to the random motion of electrons due to thermal energy. It is the average speed of electrons in a conductor due to their thermal agitation at a given temperature.

### Step 3: Analyze Conditions Without External Voltage
When no external voltage is applied to a conductor, the drift velocity is zero. This is because the electrons are in random motion and do not experience a net displacement in any direction. They move randomly and return to their original positions, resulting in no overall drift.

### Step 4: Analyze Conditions With External Voltage
When a voltage is applied, the electrons gain a net drift velocity in the direction opposite to the electric field. However, even in this case, the thermal velocity remains significant as it represents the random motion of the electrons.

### Step 5: Relationship Between Drift and Thermal Velocity
The relationship can be expressed as:
- Drift velocity (\(v_d\)) is much smaller than thermal velocity (\(v_t\)) in a conductor under normal conditions. Mathematically, this can be represented as:
\[ v_d \ll v_t \]

### Conclusion
In summary, the drift velocity of electrons is influenced by the application of an electric field, while thermal velocity is due to random thermal motion. In the absence of an electric field, both drift velocity and net displacement are zero, while thermal velocity remains non-zero due to thermal agitation.