To determine the emergent velocity of a charged ion moving through combined electric and magnetic fields, we need to analyze the forces acting on the ion and how they relate to its motion. ### Correct Option: A. E/B ### Step-by-Step Explanation: 1. **Understanding the Forces**: - When a charged particle (like an ion) moves through an electric field (E) and a magnetic field (B), it experiences two forces: - **Electric Force (F_E)**: This force is given by the equation: \[ F_E = qE \] where \( q \) is the charge of the ion and \( E \) is the electric field strength. - **Magnetic Force (F_B)**: This force is given by the equation: \[ F_B = qvB \sin(\theta) \] where \( v \) is the velocity of the ion, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the velocity vector and the magnetic field vector. For maximum force, we consider \( \theta = 90^\circ \) (i.e., the velocity is perpendicular to the magnetic field), simplifying the equation to: \[ F_B = qvB \] 2. **Condition for Uniform Motion**: - For the ion to move with a constant velocity (i.e., not accelerating), the net force acting on it must be zero. This means that the electric force must balance the magnetic force: \[ F_E = F_B \] - Substituting the expressions for the forces, we have: \[ qE = qvB \] 3. **Solving for Velocity**: - We can cancel \( q \) from both sides (assuming \( q \neq 0 \)): \[ E = vB \] - Rearranging this equation to solve for the velocity \( v \): \[ v = \frac{E}{B} \] ### Why Option A is Correct: - The derived formula \( v = \frac{E}{B} \) directly corresponds to option A. This indicates that the emergent velocity of the ion is directly proportional to the electric field strength and inversely proportional to the magnetic field strength. ### Why Other Options are Incorrect: - **Option B: EB**: This option suggests that the velocity is the product of the electric field and the magnetic field, which does not align with the derived relationship. The velocity cannot be expressed as a product of these two fields. - **Option C: B/E**: This option implies that the velocity is inversely proportional to the electric field and directly proportional to the magnetic field, which contradicts our derived formula. The correct relationship shows that the velocity increases with increasing electric field strength and decreases with increasing magnetic field strength. - **Option D: E - B**: This option suggests a subtraction of the electric and magnetic fields, which is not relevant in the context of forces acting on the charged ion. The forces do not combine in this manner; they must be balanced for uniform motion. ### Common Pitfalls: - **Forgetting the Direction of Forces**: It's crucial to remember that the electric and magnetic forces act in different directions. The electric force acts in the direction of the electric field, while the magnetic force acts perpendicular to both the velocity and the magnetic field. - **Assuming Constant Charge**: The derivation assumes that the charge of the ion is constant and non-zero. If the charge were zero, there would be no force acting on the ion. - **Neglecting the Angle**: The derivation assumes that the angle between the velocity and magnetic field is 90 degrees for maximum force. If the angle is different, the magnetic force would be less than \( qvB \). ### Revision Summary: - The emergent velocity of a charged ion in combined electric and magnetic fields is given by \( v = \frac{E}{B} \). - The electric force must balance the magnetic force for the ion to move with constant velocity. - The correct answer is option A, as it accurately reflects the relationship derived from the forces acting on the ion. - Be cautious of the direction of forces and the conditions under which the derived formula applies.