A constant voltage is applied across a wire of constant length then how drift velocity of electrons depends on area of cross-section of wire.

Explanation

To determine how the drift velocity of electrons depends on the area of cross-section of a wire when a constant voltage is applied across it, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of 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 given by the equation:
\[ V_d = \mu E \]
where \( \mu \) is the mobility of the electrons, and \( E \) is the electric field.

2. **Relate Electric Field to Voltage**:
The electric field (\( E \)) in the wire can be expressed in terms of the voltage (\( V \)) applied across the wire and the length (\( L \)) of the wire:
\[ E = \frac{V}{L} \]

3. **Substitute Electric Field into Drift Velocity Equation**:
By substituting the expression for electric field into the drift velocity equation, we get:
\[ V_d = \mu \left(\frac{V}{L}\right) \]

4. **Consider the Mobility of Electrons**:
The mobility (\( \mu \)) of free electrons is a constant that depends on the material of the wire and does not depend on the area of cross-section.

5. **Analyze the Dependence on Area**:
The current (\( I \)) flowing through the wire can be expressed as:
\[ I = n A q V_d \]
where \( n \) is the number density of charge carriers, \( A \) is the area of cross-section, and \( q \) is the charge of an electron. Rearranging gives:
\[ V_d = \frac{I}{n A q} \]
However, since we are applying a constant voltage and the length of the wire is constant, the current \( I \) is also constant for a given voltage.

6. **Conclusion**:
From the above equations, we can see that while the drift velocity can be expressed in terms of the area of cross-section, it does not depend on the area when a constant voltage is applied across a wire of constant length. Thus, the drift velocity remains independent of the area of cross-section.

### Final Answer:
The drift velocity of electrons does not depend on the area of cross-section of the wire when a constant voltage is applied across a wire of constant length.

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