Moving Charges And Magnetism

The magnetic moment of a current-carrying loop can be found by breaking the loop into sub-loops because of the superposition principle, where the total magnetic moment is the vector sum of the moments of the sub-loops. This approach simplifies calculations, especially for loops with complex shapes, by allowing the use of simpler geometric sub-loops. The total magnetic moment is given by , where the area vector of the entire loop is the sum of the area vectors of the sub-loops, since the same current flows through all parts.$ $

The motion of a charged particle in a uniform magnetic field can be decomposed into two independent motions:Motion parallel to the magnetic field, which is uniform. This component determines the pitch ( ) of the helical trajectory.Motion perpendicular to the magnetic field, which is circular. This component determines the radius ( ) of the helical trajectory.Let the given velocity be and the magnetic field be .First, we need to find the components of the velocity vector parallel ( ) and perpendicular ( ) to the magnetic field.1. Calculate the magnitude of the magnetic field and velocity: 2. Calculate the component of velocity parallel to the magnetic field ( ):The dot product of and is: The magnitude of the velocity component parallel to is : 3. Calculate the component of velocity perpendicular to the magnetic field ( ):We know that .So, 4. Calculate the radius of the helical trajectory ( ):The magnetic force provides the centripetal force for the circular motion. Substitute the values of and : 5. Calculate the pitch of the helical trajectory ( ):The pitch is the distance traveled along the magnetic field direction in one period of circular motion.First, calculate the period of the circular motion: Now, calculate the pitch : Comparing our calculated values with the given options: These values match Option A.The final answer is