The correct option for the direction of the magnetic force acting on a charged particle moving with a velocity \( v \) perpendicular to a uniform magnetic field \( B \) is: **C. Perpendicular to both the velocity and the magnetic field.** ### Detailed Explanation 1. **Understanding the Context**: - When a charged particle (like an electron or proton) moves through a magnetic field, it experiences a magnetic force. This force is a result of the interaction between the charge of the particle and the magnetic field. 2. **The Right-Hand Rule**: - The right-hand rule is a mnemonic used to determine the direction of the magnetic force on a charged particle. Here’s how to apply it: - Point your right thumb in the direction of the velocity \( v \) of the charged particle. - Point your fingers in the direction of the magnetic field \( B \). - The direction your palm pushes (or the direction your palm faces) will indicate the direction of the magnetic force \( F \) acting on a positive charge. For a negative charge, the force will be in the opposite direction. 3. **Mathematical Representation**: - The magnetic force \( F \) on a charged particle can be calculated using the formula: \[ F = q(v \times B) \] where: - \( F \) is the magnetic force, - \( q \) is the charge of the particle, - \( v \) is the velocity vector of the particle, - \( B \) is the magnetic field vector, - \( \times \) denotes the cross product. - The cross product \( v \times B \) results in a vector that is perpendicular to both \( v \) and \( B \). This is a fundamental property of the cross product in vector mathematics. 4. **Why Option C is Correct**: - Since the magnetic force is given by the cross product of the velocity and the magnetic field, it is inherently perpendicular to both vectors. Therefore, the correct answer is that the magnetic force acts **perpendicular to both the velocity and the magnetic field**. ### Why the Other Options are Incorrect - **Option A: In the direction of the magnetic field**: - This option is incorrect because the magnetic force does not act in the direction of the magnetic field. Instead, it is perpendicular to the field. - **Option B: Opposite to the direction of the particle's velocity**: - This option is also incorrect. The magnetic force does not act directly opposite to the velocity of the particle. Instead, it acts perpendicular to the velocity, which means it can change the direction of the particle's motion but not its speed. - **Option D: In the same direction as the particle's velocity**: - This option is incorrect as well. The magnetic force cannot act in the same direction as the velocity because it is always perpendicular to the velocity vector when the particle is moving in a magnetic field. ### Common Pitfalls - **Confusing the Direction of Force**: Students often confuse the direction of the magnetic force with the direction of the velocity or the magnetic field. Remember, the force is always perpendicular to both. - **Forgetting the Right-Hand Rule**: Not applying the right-hand rule correctly can lead to mistakes in determining the direction of the force. - **Neglecting Charge Sign**: The direction of the force is affected by the sign of the charge. For negative charges, the force direction is opposite to what the right-hand rule indicates. ### Revision Summary - The magnetic force on a charged particle moving in a magnetic field is given by \( F = q(v \times B) \). - Use the right-hand rule: Thumb for velocity, fingers for magnetic field, palm for force direction. - The magnetic force is always **perpendicular** to both the velocity and the magnetic field. - Remember that the direction of the force depends on the sign of the charge: positive charges follow the right-hand rule, while negative charges do the opposite.