Let's break down the problem step by step.

Given Information:

1. Speed of electron: $5.2 \times 10^5 \, \text{m/s}$ in the positive $x$-direction.
2. Zero magnetic force when moving in the positive $x$-direction.
3. Force experienced: When the electron moves in the positive $y$-direction, it experiences a magnetic force of $4.1 \times 10^{-13} \, \text{N}$ in the positive $z$-direction.

Key Concepts:

The force on a charged particle moving in a magnetic field is given by the Lorentz force equation:

$\mathbf{F} = q (\mathbf{v} \times \mathbf{B})$

Where:

• $\mathbf{F}$ is the force,
• $q$ is the charge of the particle (for an electron, $q = -1.6 \times 10^{-19} \, \text{C}$),
• $\mathbf{v}$ is the velocity of the particle,
• $\mathbf{B}$ is the magnetic field, and
• $\times$ denotes the cross product.

Analysis:

1. No force in the $x$-direction: When the electron is moving in the positive $x$-direction, there is no force. This implies that the magnetic field $\mathbf{B}$ has no component in the $y$- or $z$-directions because $\mathbf{v} \times \mathbf{B} = 0$ when $\mathbf{v} = v_x \hat{i}$.

   Hence, $\mathbf{B}$ must be in the $x$-direction.

2. Force in the $z$-direction: When the electron moves in the $y$-direction, it experiences a force in the positive $z$-direction. The cross product $\mathbf{v} \times \mathbf{B}$ gives a force in the $z$-direction if $\mathbf{v}$ is in the $y$-direction and $\mathbf{B}$ is in the negative $x$-direction.

Conclusion:

The magnetic field is in the negative $x$-direction, i.e., $-\hat{i}$.

Thus, the correct answer is $-\hat{i}$, which corresponds to your selection in the image.

Do you need further details or have any questions?

Here are 5 related questions for further understanding:

1. What happens if the magnetic field is along the positive $x$-direction?
2. How would the force change if the charge of the particle was positive instead of negative?
3. Can you calculate the magnitude of the magnetic field based on the given data?
4. How would the scenario change if the particle was moving in a different direction, like the negative $y$-axis?
5. What is the relationship between the direction of velocity, magnetic field, and force?

Tip: Always apply the right-hand rule for cross products involving the velocity and magnetic field to determine the direction of the force on a charged particle.