MEC BYU 7

Solution: Trajectory in a Time-Varying Magnetic Field

Analysis: The magnetic field acts as a square wave, switching direction every time interval $T$.

• 0 to T: Field is $+B$. Particle moves in a circle (e.g., clockwise).
• T to 2T: Field is $-B$. Force reverses. Particle moves in a circle of the same radius but opposite sense (counter-clockwise).

For the path to be closed and resemble a “digit 8”, the particle must complete an integer number of full circles (or a specific symmetry) within the time $T$ before the field flips.

Condition:
The time interval $T$ must correspond to an integer multiple of the time period of revolution $T_{cyclotron}$.

$$ T = n \times T_{cyclotron} \quad \text{where } n = 1, 2, 3 \dots $$

The cyclotron period is given by: $$ T_{cyclotron} = \frac{2\pi m}{qB} $$

Substitute and solve for charge $q$: $$ T = n \left( \frac{2\pi m}{qB} \right) $$