A cylindrical copper conductor AB length `L` areaa of cross-section a has large number of free electrons which at mean temperature move at random within the body of the conductor like the molecules of a gas. The average thermal motion at room temperature is of the enter of `10^(5)ms^(-1)` where a potential difference `V` is applied free electronic in the condictior experience , the free electrons in the conductor experience force and are accelerated towards the positive emf of the condutor on their gained kinetic energy After each collision the free electronic are angle acceleration due of the electric field , towards the positive end the conductor and next collision with the ions/atoms of the electrons The average speed of the free electrons with which they drift toward the positive and of the conductor under the effect of applied electric field is called drift of the electrons
The drift speed of the electrons depends on

Options

A. dimension of the conductor
B. number density of free electrons in the conductor
C. both (a) and (b)
D. none of these above

Explanation

To determine the factors on which the drift speed of electrons in a cylindrical copper conductor depends, we can start by analyzing the concept of drift velocity.

### Step-by-Step Solution:

1. **Understanding Drift Velocity**:
The drift velocity (\(V_d\)) of free electrons in a conductor is the average velocity that an electron attains due to an electric field. It is a result of the superposition of the random thermal motion of electrons and the directed motion caused by the electric field.

**Hint**: Recall that drift velocity is different from thermal velocity; it is the net velocity in the direction of the electric field.

2. **Formula for Drift Velocity**:
The drift velocity can be expressed using the formula:
\[ V_d = \frac{E \cdot \tau}{m} \]
where:
- \(E\) is the electric field strength,
- \(\tau\) is the relaxation time (the average time between collisions of electrons with atoms),
- \(m\) is the mass of the electron.

**Hint**: Identify the components of the formula and their physical significance.

3. **Factors Affecting Drift Velocity**:
- **Electric Field (E)**: The drift velocity is directly proportional to the electric field. A stronger electric field results in a higher drift velocity.
- **Relaxation Time (\(\tau\))**: This depends on the material properties and temperature. A longer relaxation time means that electrons can accelerate more before colliding with the lattice ions, leading to a higher drift velocity.
- **Mass of the Electron (m)**: The drift velocity is inversely proportional to the mass of the electron. However, since the mass of the electron is a constant, it does not change with the dimensions of the conductor.

**Hint**: Think about how each factor influences the motion of the electrons.

4. **Dependence on Dimensions and Electron Density**:
- The dimensions of the conductor (length \(L\) and cross-sectional area \(A\)) do not directly affect the drift velocity itself. However, they do influence the overall current and resistance.
- The number density of free electrons affects the current but does not directly change the drift velocity formula.

**Hint**: Consider how the physical properties of the conductor relate to the drift velocity.

5. **Conclusion**:
From the analysis, we conclude that the drift speed of electrons depends primarily on the electric field and relaxation time, rather than the dimensions of the conductor or the number density of free electrons. Therefore, the correct answer to the question is that the drift speed does not depend on the dimensions of the conductor or the number density of free electrons.

**Final Answer**: The drift speed of the electrons does not depend on the dimensions of the conductor or the number density of free electrons.