In the rectangle in the drawing, a charge is to be placed at the empty corner to make the net force on the charge at corner A point along the vertical direction. What charge (magnitude and algebraic sign) must be placed at the empty corner?
To determine the charge that needs to be placed at the empty corner, we need to analyze the forces acting on the charge at corner A. From the given information, we know that we want the net force on the charge at corner A to point along the vertical direction.
First, let's label the charges in the rectangle corners: A, B, C, and D. Let's assume that the charges at corners B, C, and D are +Q, -Q, and +Q, respectively, where Q represents the magnitude of charge. We do not know the charge at corner A yet.
Now, let's consider the forces acting on the charge at corner A. We have the following forces:
1. The force F1 between the charge at corner A and the charge at corner B:
- This force is attractive since opposite charges attract.
- The magnitude of this force depends on the distance between the charges and the magnitude of the charges.
2. The force F2 between the charge at corner A and the charge at corner C:
- This force is repulsive since like charges repel.
- The magnitude of this force depends on the distance between the charges and the magnitude of the charges.
3. The force F3 between the charge at corner A and the charge at corner D:
- This force is attractive since opposite charges attract.
- The magnitude of this force depends on the distance between the charges and the magnitude of the charges.
Now, since we want the net force on the charge at corner A to point along the vertical direction, the magnitudes of the vertical forces must be equal. This means that the magnitude of force F1 must be equal to the magnitude of force F3.
To achieve this, we need to make sure that the magnitudes of the charges at corners A and D are equal. In other words, the charge at corner A should also be +Q.
Therefore, to make the net force on the charge at corner A point along the vertical direction, the charge at the empty corner should be +Q.