The correct option is **C. Perpendicular to both the velocity of the particle and the magnetic field**. ### Detailed Explanation: To understand the direction of the magnetic force acting on a charged particle moving in a magnetic field, we can use the **Lorentz force law**, which states that the magnetic force (\( \mathbf{F} \)) on a charged particle is given by the equation: \[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \] Where: - \( \mathbf{F} \) is the magnetic force, - \( q \) is the charge of the particle, - \( \mathbf{v} \) is the velocity vector of the particle, - \( \mathbf{B} \) is the magnetic field vector, - \( \times \) denotes the cross product. #### Step-by-Step Analysis: 1. **Understanding the Cross Product**: The cross product of two vectors results in a third vector that is perpendicular to the plane formed by the two original vectors. In this case, the vectors are the velocity of the particle (\( \mathbf{v} \)) and the magnetic field (\( \mathbf{B} \)). 2. **Direction of the Force**: Since the particle is positively charged, the direction of the magnetic force will be determined by the right-hand rule. According to this rule: - Point your right thumb in the direction of the velocity vector (\( \mathbf{v} \)). - Curl your fingers in the direction of the magnetic field vector (\( \mathbf{B} \)). - Your palm will then point in the direction of the magnetic force (\( \mathbf{F} \)) acting on the positively charged particle. 3. **Perpendicular Motion**: The problem states that the particle is moving **perpendicular** to the magnetic field lines. This means that the angle between the velocity vector and the magnetic field vector is \( 90^\circ \). The sine of \( 90^\circ \) is 1, which maximizes the force according to the formula: \[ F = qvB \sin(\theta) \] Where \( \theta \) is the angle between \( \mathbf{v} \) and \( \mathbf{B} \). Since \( \theta = 90^\circ \), the force is at its maximum and is directed perpendicular to both the velocity and the magnetic field. ### Why Other Options Are Incorrect: - **Option A: In the same direction as the particle's velocity**: This is incorrect because the magnetic force is not in the same direction as the velocity. The force acts perpendicular to the motion of the particle, causing it to change direction rather than speed. - **Option B: Opposite to the direction of the magnetic field**: This is also incorrect. The magnetic force does not act in the direction opposite to the magnetic field. Instead, it acts perpendicular to both the magnetic field and the velocity of the charged particle. - **Option D: In the direction of the magnetic field lines**: This option is incorrect as well. The magnetic force does not align with the magnetic field lines; it is always perpendicular to them when the charged particle is moving perpendicular to the field. ### Summary of Key Points: - The magnetic force on a charged particle moving in a magnetic field is given by the Lorentz force law. - The direction of the magnetic force is determined by the right-hand rule and is always perpendicular to both the velocity of the particle and the magnetic field. - For a positively charged particle moving perpendicular to the magnetic field, the force is maximized and acts perpendicular to both vectors. - Understanding the cross product and the right-hand rule is crucial for determining the direction of the magnetic force. This thorough understanding will help you tackle similar questions in your physics exams effectively!