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Consider the arrangement of four positive charges shown, each being equidistant from the origin. If a small positive charge q is placed at the origin, the net resultant force acting on q will be in what quadrant?

Q01_P03
None, because it experiences no net force.
B
C
D
A

1 answer

To determine the direction of the net force acting on a small positive charge \( q \) placed at the origin due to four positive charges arranged around it, we need to analyze the forces exerted by each charge on the charge at the origin.

Let's assume the charges are located at the following positions in the Cartesian coordinate system:

  • Charge \( +Q_1 \) in the first quadrant (positive x and y).
  • Charge \( +Q_2 \) in the second quadrant (negative x and positive y).
  • Charge \( +Q_3 \) in the third quadrant (negative x and y).
  • Charge \( +Q_4 \) in the fourth quadrant (positive x and negative y).

Since all four charges are positive, each of them will exert a repulsive force on the point charge \( q \) at the origin.

  • The force from charge \( +Q_1 \) will push \( q \) away towards the negative x and negative y (quadrant III).
  • The force from charge \( +Q_2 \) will push \( q \) away towards the negative x and positive y (quadrant II).
  • The force from charge \( +Q_3 \) will push \( q \) away towards the positive x and positive y (quadrant I).
  • The force from charge \( +Q_4 \) will push \( q \) away towards the positive x and negative y (quadrant IV).

To analyze the net force, each force can be broken down into its x and y components:

  1. The forces in the x-direction from \( Q_1 \) and \( Q_3 \) act towards the negative x direction, while \( Q_2 \) and \( Q_4 \) act towards the positive x direction.
  2. The forces in the y-direction from \( Q_1 \) and \( Q_4 \) act towards the positive y direction, while \( Q_2 \) and \( Q_3 \) act towards the negative y direction.

Since the charges are symmetrically placed around the origin, the forces in the x-axis and y-axis will not cancel each other out completely, leading to a distinctive net force. The resultant direction will be dictated by the overall balance of these forces.

Assuming equal magnitudes and symmetry, the net force will be toward the first quadrant (where forces point outward from the origin) as the magnitude of forces from \( Q_1 \) and \( Q_4 \) will balance those from \( Q_2 \) and \( Q_3 \).

Thus, the net resultant force acting on \( q \) will be in Quadrant I.