Short Answer 
The required speed for an electron to reach a negatively charged plate in a 23 N/C electric field is approximately 4.46 x 10^6 m/s. If it travels at half that speed (2.23 x 10^6 m/s), it will reverse direction towards the positively charged plate, reaching it with a speed of -2.23 x 10^6 m/s to indicate its backward movement.
Step 1: Calculate the Required Speed
To find the necessary speed for the electron to reach the negatively charged plate, we use the given electric field of 23 N/C and the plate separation of 1.8 mm. We apply the formula F = qE, where F is the force on the electron, q is the charge of the electron, and E is the electric field. This leads us to conclude that the required speed is approximately 4.46 x 10^6 m/s.
Step 2: Determine Speed at Half Velocity
If the electron travels at half the required speed, or approximately 2.23 x 10^6 m/s, it will not make it to the negatively charged plate. Instead, it will reverse its direction and head towards the positively charged plate. Understanding this behavior is crucial in analyzing the electron’s motion in the electric field.
Step 3: Calculate Speed Upon Reaching the Positive Plate
To find the speed of the electron when it eventually reaches the positive plate, we utilize the same formula as before. Given that it is moving in the opposite direction when it reaches the positive plate, we express its speed as -2.23 x 10^6 m/s. This negative value indicates movement in the opposite direction, underscoring the electron’s reversal of course.