To analyze the given statements about the behavior of free electrons in a metallic conductor when an electric field is applied, we can break down the reasoning step by step.
### Step-by-Step Solution:
1. **Understanding Free Electrons in a Conductor**:
- In a metallic conductor, there are free electrons that can move under the influence of an electric field. These electrons are not bound to any particular atom and can drift through the lattice of positively charged metal ions.
**Hint**: Recall that metals have a sea of free electrons that allow them to conduct electricity.
2. **Effect of Electric Field**:
- When an electric field is applied, the free electrons experience a force due to the electric field. The direction of this force is opposite to the direction of the electric field since electrons are negatively charged.
**Hint**: Remember the relationship between electric field (E), charge (q), and force (F = qE).
3. **Initial Acceleration**:
- Initially, the electrons will accelerate due to the force exerted by the electric field. The acceleration can be expressed as \( a = \frac{F}{m} = \frac{eE}{m} \), where \( e \) is the charge of the electron and \( m \) is its mass.
**Hint**: Use Newton's second law to relate force, mass, and acceleration.
4. **Collisions with Atoms**:
- As the electrons accelerate, they will collide with the fixed positively charged ions in the conductor. These collisions are frequent and random, which impede the continuous acceleration of the electrons.
**Hint**: Consider how collisions affect the motion of particles in a medium.
5. **Drift Velocity**:
- After many collisions, the electrons reach a constant average velocity known as the drift velocity. This drift velocity is not zero, but it is constant because the electrons are constantly colliding with the atoms, which balances the acceleration due to the electric field.
**Hint**: Drift velocity is a key concept in understanding current flow in conductors.
6. **Evaluating Statement I**:
- Statement I claims that free electrons do not accelerate but drift at a constant speed. This is correct; while they initially accelerate, they reach a steady drift speed due to collisions.
**Hint**: Think about how drift speed relates to the net motion of electrons in a conductor.
7. **Evaluating Statement II**:
- Statement II claims that the force exerted by the electric field is balanced by the Coulomb force between electrons and protons. This is incorrect; the force is balanced by the collision force with the lattice atoms, not by the Coulomb force.
**Hint**: Distinguish between the forces acting on electrons due to collisions and those due to electric fields.
8. **Conclusion**:
- Therefore, Statement I is true, and Statement II is false. The correct answer is that Statement I is true, and Statement II is false.
### Final Answer:
- Statement I is true; Statement II is false.
