To solve the question regarding which statement is correct about the behavior of charged particles in a magnetic field, we will analyze each option step by step.
### Step-by-Step Solution:
1. **Understanding the Force on a Charged Particle in a Magnetic Field**:
- A charged particle moving in a magnetic field experiences a magnetic force given by the equation:
\[
F = q(v \times B)
\]
- Here, \( q \) is the charge, \( v \) is the velocity of the particle, and \( B \) is the magnetic field. The direction of the force is given by the right-hand rule.
**Hint**: Remember that the force is dependent on the charge, velocity, and magnetic field, and it acts perpendicular to both the velocity and the magnetic field.
2. **Analyzing the First Option**:
- The first option states that a charged particle can be accelerated by a magnetic field. Since the magnetic force acts on the particle, it can indeed cause acceleration.
- Therefore, this statement is **correct**.
**Hint**: Acceleration occurs when a net force is applied, and in this case, the magnetic force provides that net force.
3. **Analyzing the Second Option**:
- The second option claims that a charged particle cannot be accelerated by a magnetic field. This is incorrect because we established that the magnetic force can cause acceleration.
- Thus, this statement is **wrong**.
**Hint**: Recall that acceleration is possible when a force is applied, and magnetic fields can exert such forces.
4. **Analyzing the Third Option**:
- The third option suggests that the speed of the charged particle can be increased by a uniform magnetic field. However, since the force from the magnetic field is always perpendicular to the velocity, it can change the direction of the particle but not its speed.
- Therefore, this statement is **wrong**.
**Hint**: Think about how the work done by a force affects speed; if the force is perpendicular, no work is done on the particle.
5. **Analyzing the Fourth Option**:
- The fourth option states that the speed of the charged particle can be increased by a non-uniform magnetic field. Similar to the third option, the magnetic force remains perpendicular to the velocity, regardless of whether the magnetic field is uniform or non-uniform.
- Hence, this statement is also **wrong**.
**Hint**: The nature of the magnetic field (uniform or non-uniform) does not change the fact that the force is perpendicular to the velocity.
### Conclusion:
After analyzing all the options, we conclude that the only correct statement is the first option: **A charged particle can be accelerated by a magnetic field**.
