Understanding the Motion of a Charged Particle in a Magnetic Field
When a charged particle with mass \( m \) and charge \( Q \) enters a uniform magnetic field with speed \( v \) at an angle \( \theta \), it experiences a magnetic force given by the Lorentz force equation:

Magnetic Force
- The magnetic force \( \mathbf{F} \) acting on the particle is given by:
\[
\mathbf{F} = Q(\mathbf{v} \times \mathbf{B})
\]
- Here, \( \mathbf{B} \) is the magnetic field vector.

Force Direction and Average Force Calculation
- The magnetic force is always perpendicular to both the velocity of the particle and the magnetic field.
- This results in the particle undergoing circular motion in the plane perpendicular to the magnetic field.

Time Duration of Interest
- The time duration \( T = \frac{2\pi m}{Q B} \) is the time it takes for the particle to complete one full circular path in the magnetic field.

Average Force Over One Cycle
- Over one complete cycle (time \( T \)), the particle returns to its original position with the same velocity, hence:
- The net displacement is zero.
- The average force \( \langle \mathbf{F} \rangle \) over this period is zero since the particle does not experience a net change in momentum.

Conclusion
- Therefore, the average force acting on the particle during the time \( \frac{2\pi m}{Q B} \) is indeed zero.
- This conclusion holds true irrespective of the angle \( \theta \) at which the particle is thrown, as long as it remains in the magnetic field.
In summary, the magnetic field causes the particle to move in a circular path, resulting in a zero average force over one complete cycle.