Short Answer 
The radius ( r ) of a charged particle in a magnetic field is calculated using the formula ( r = mv / qB ), where mass, velocity, charge, and magnetic field strength are key factors. When the mass is doubled and the charge halved, the new radius ( r’ ) becomes ( 4r ), indicating that the path’s size increases proportionally with these changes.
Step 1: Understanding the Radius Formula
The radius ( r ) of a charged particle moving in a magnetic field is given by the formula:
- r = mv / qB
This formula shows that the radius (r) depends on the mass (m), velocity (v), charge (q), and magnetic field strength (B). Each of these factors plays a critical role in determining how large the circular path will be.
Step 2: Analyzing Changes in Mass and Charge
If the particle’s mass is increased to 2m and its charge decreased to q/2, we need to calculate the new radius ( r’ ) using the same formula:
- Insert values: r’ = (2m)v / (q/2)B
- Simplify the equation: r’ = (2mv * 2) / qB
- This results in: r’ = 4mv / qB
Thus, the new equation for the radius reflects how the changes in mass and charge alter the overall dynamics of the motion.
Step 3: Relating New Radius to Original Radius
Since we identified that the original radius ( r ) can be stated as ( r = mv / qB ), we can now express the new radius ( r’ ) in terms of the original radius:
- r’ = 4(mv / qB)
- This implies: r’ = 4 ‚à öo r
This demonstrates that after altering the mass and charge of the particle, the new radius of the path becomes ( 4 times r ), indicating a proportional relationship based on the changes made.