91Ó°ÊÓ

When a positively charged particle is released in a region with an electromagnetic wave, it experiences an electric force due to the electric field component of the wave. This force is calculated using the equation \( \mathbf{F}_e = q \mathbf{E} \), where \( q \) is the charge of the particle and \( \mathbf{E} \) represents the electric field vector. In our scenario, the electric field is given by \( \mathbf{E} = \hat{\mathbf{x}} \tilde{E} \sin \omega t \).

This means that the force is directed along the x-axis and varies sinusoidally with time. The particle begins to move due to this force, and its acceleration can be analyzed using Newton's second law, \( F = ma \). Substituting for the electric force, we have

- \( m \frac{d^2 x}{dt^2} = q \tilde{E} \sin \omega t \).

By integrating the equation twice, we find the position \( x(t) \) of the particle as a function of time:

- \( x(t) = \frac{q \tilde{E}}{m \omega^2} \sin \omega t \),

assuming that the particle starts from rest at the origin. This expression describes how the particle oscillates along the x-axis under the influence of the electric field.

In addition to the electric force, the charged particle is also subjected to a magnetic force due to the magnetic field component of the electromagnetic wave. This force is a result of the interaction between the particle's velocity and the magnetic field, described by \( \mathbf{F}_b = q \mathbf{v} \times \mathbf{B} \).

Given the magnetic field \( \mathbf{B} = \hat{\mathbf{y}} \tilde{B} \sin \omega t \) and the velocity of the particle \( v_x(t) = \frac{q \tilde{E}}{m \omega} \sin \omega t \), we find the magnetic force:

- \( \mathbf{F}_b = -\hat{\mathbf{z}} \frac{q^2 \tilde{E} \tilde{B} \sin^2 \omega t}{m \omega} \).

This force acts along the z-axis and its direction changes with the square of the sine of the wave's argument.

Although it might seem complex, breaking it down reveals that the negatively directed z-component is due to the cross product nature and the sinusoidal time variance of both velocity and magnetic field which causes oscillations in the z-direction.

In the described problem, it's noted that the particle has a relatively large \( m/q \) ratio, ensuring non-relativistic dynamics. This implies that the particle’s speed is much less than the speed of light, making relativistic effects negligible.

Despite the complexity of the motion under electromagnetic influence, the particle’s dynamics remain within limits that can be managed through classical mechanics. In this case, the non-relativistic description simplifies the problem to solving ordinary differential equations of motion.

For the z-component of motion, the particle senses a force due to the magnetic interaction. Applying Newton's second law again results in:

- \( m \frac{d^2 z}{dt^2} = -\frac{q^2 \tilde{E} \tilde{B} \sin^2 \omega t}{m \omega} \).

Integration shows that the motion in the z-direction is bounded, resulting in a sinusoidally confined oscillation described by:

- \( z(t) = -\frac{q^2 \tilde{E} \tilde{B}}{8 m^2 \omega^3} \left( 3\omega t - \sin 2\omega t \right) \).

The existence of \( z_{\max} \) guarantees that the motion is constrained within specific limits, calculated to be \( z_{\max} = 1.101\left(\frac{q^{2} \tilde{E} \tilde{B}}{m^{2} \omega^{3}}\right) \). Thus, the particle does not travel indefinitely but oscillates within a predictable and bounded range.