Final answer:
The direction of the magnetic field causing an electron to follow a semicircular path is found using the left-hand rule due to the electron's negative charge, and it must be perpendicular to the velocity of the electron and into the page.
Step-by-step explanation:
To determine the direction of the magnetic field necessary for an electron to follow a semicircular path from point A to point B, we use the right-hand rule for magnetic force. Considering that the electron has a negative charge, we must modify the right-hand rule and use the left hand instead. Placing the thumb in the direction of the electron's velocity (which is initially the +x-direction) and the fingers in the direction of the desired path (from A to B, a semicircle), the palm of the hand will face the direction of the required magnetic field. Since the electron's path is semicircular, the magnetic field must be perpendicular to the plane of the semicircular path. As the problem states the field is directed into the page, this aligns with our analysis using the left-hand rule.
The formula F = qvB sin θ shows that the magnetic force F, acting on a charge q moving at velocity v in a magnetic field B, is perpendicular to both the velocity and the magnetic field when the charge moves perpendicular (θ = 90 degrees) to the magnetic field. This force provides the centripetal force necessary for the electron to follow a circular path.