Answer: The magnetic field has a magnitude of approximately 1.96 * 10^-4 T and points in the positive x direction. The magnetic field vector is B = (1.96 * 10^-4, 0, 0) T.
Step-by-step explanation:
To find the direction and magnitude of the magnetic field, we'll use the formula for the magnetic force acting on a moving charged particle:
F = q * (v x B)
where F is the magnetic force, q is the charge of the particle, v is its velocity, and B is the magnetic field. The cross product (v x B) implies that the force is perpendicular to both the velocity of the particle and the magnetic field.
Since the electron experiences zero magnetic force when it moves in the positive x direction, we can conclude that the magnetic field is either parallel or antiparallel to the x-axis.
Now let's analyze the situation when the electron moves in the positive y direction and experiences a force in the negative z direction. The electron charge is negative, with q = -1.6 * 10^-19 C. The velocity vector is v = (0, 4.2 * 10^5, 0) m/s, and the force vector is F = (0, 0, -2.0 * 10^-13) N.
Using the formula for magnetic force, we have:
F = q * (v x B)
Now, we need to find the cross product (v x B):
v x B = (0, 4.2 * 10^5, 0) x (Bx, 0, 0)
Since the cross product is orthogonal to both vectors, and we know that the force is in the negative z direction, we can conclude that the magnetic field must be in the positive x direction (parallel to the x-axis). So, the magnetic field vector is B = (Bx, 0, 0).
Now we can rewrite the force equation:
(0, 0, -2.0 * 10^-13) = -1.6 * 10^-19 * (0, 0, 4.2 * 10^5 * Bx)
To solve for Bx, we can focus on the z component of the force equation:
-2.0 * 10^-13 = -1.6 * 10^-19 * (4.2 * 10^5 * Bx)
Dividing both sides by (-1.6 * 10^-19 * 4.2 * 10^5):
Bx = (2.0 * 10^-13) / (-1.6 * 10^-19 * 4.2 * 10^5)
Bx ≈ 1.96 * 10^-4 T
Thus, the magnetic field has a magnitude of approximately 1.96 * 10^-4 T and points in the positive x direction. The magnetic field vector is B = (1.96 * 10^-4, 0, 0) T.