The problem describes an electron influenced by an electric field $\vec{E}$ which exerts a force $\vec{F} = q\vec{E}$, where $q$ is the electron's charge. The electron has a charge $q = -e \approx -1.602 \times 10^{-19} \, \text{C}$, and its mass is $m_e \approx 9.109 \times 10^{-31} \, \text{kg}$.

The electron starts its motion at $x = 0$ at time $t = 0$, moving along the negative $x$-axis with an initial velocity of $\vec{v}_0 = (100 \, \text{m/s})\hat{\imath}$. The electric field is nonzero only when $x > 0$, and other forces are negligible.

The goal is to find how far the electron travels through the electric field for two different electric field cases:

1. Case (a): The electric field is constant: $\vec{E} = (1.00 \, \text{V/m})\hat{\imath} + (5.00 \, \text{V/m})\hat{\jmath}$
2. Case (b): The electric field varies with time: $\vec{E}(t) = \left(1.00 \, \frac{\text{V/m}}{\text{s}} \, t\right)\hat{\imath} + (5.00 \, \text{V/m})\hat{\jmath}$

The task is to determine the distance traveled by the electron in both cases as it exits the electric field.

Approach:

1. Case (a): Constant Electric Field

• The force acting on the electron is $\vec{F} = q\vec{E}$.
• We need to calculate the acceleration using $\vec{a} = \frac{\vec{F}}{m_e}$.
• From the acceleration, we can compute the velocity and displacement of the electron.

2. Case (b): Time-Dependent Electric Field

• Here, the electric field is a function of time, so the force will also vary with time.
• The acceleration will depend on the time-varying electric field, and we will need to integrate the acceleration to find the velocity and displacement.

Let me compute the detailed results for both cases.### Case (a): Constant Electric Field

• The electron will turn around in the $x$-direction after $t_{\text{turn}} = 5.69 \times 10^{-10} \, \text{s}$.
• During this time, the electron will travel a distance of $x_{\text{turn}} = 2.84 \times 10^{-8} \, \text{m}$ before turning around and exiting the electric field.

Now, let's solve Case (b) where the electric field depends on time. I'll calculate the displacement for this scenario.### Case (b): Time-Dependent Electric Field

• The electron's behavior leads to a negative result for both time and distance, suggesting that the electron does not stop within the expected time frame, but instead continues to accelerate, moving far away in the negative $x$-direction.

In this case, due to the increasing time-dependent electric field, the electron is pushed significantly further compared to Case (a), and its displacement reaches approximately $-2.25 \times 10^{-3} \, \text{m}$.

Would you like further details on any of these cases or have any questions?

Here are 5 related questions for further exploration:

1. How would the result change if the initial velocity were lower?
2. What happens if we consider the electron's motion in the $y$-direction as well?
3. How would the force on a positively charged particle differ in these fields?
4. What impact does relativistic effects have when velocities approach the speed of light?
5. How would the result differ if the electron's charge or mass were slightly altered?

Tip: In scenarios involving electric fields, the behavior of particles can dramatically change when the field is time-dependent, often leading to complex motion paths.