A charge is to be placed at the empty corner to make the net force at corner A point along the vertical direction. What charge (magnitude and algebraic sign) must be placed at the empty corner if the three charges have the same charge of +7.63 μC?
To determine the charge needed at the empty corner, we need to analyze the forces on the charges at corner A. Let's break down the steps:
1. Understand the setup: We have a situation where three charges are placed at the corners of a triangle, and there is an empty corner. The three charges all have the same charge of +7.63 μC.
2. Identify the forces: The net force on a charge will be the vector sum of forces due to the other charges. In this case, we need to consider the forces on the charge at corner A.
3. Determine the forces: Since the charges at corners B and C are identical (+7.63 μC), they will exert forces of the same magnitude on the charge at corner A. These forces will be directed along the lines connecting the charges.
4. Resolve the forces: The net force at corner A needs to point along the vertical direction. Therefore, we need to determine the algebraic sum of the forces in the vertical direction and set it equal to zero.
5. Apply Coulomb's law: Coulomb's law states that the force between two charges is proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. We can use this equation to calculate the magnitudes of the forces.
6. Set up the equations: Let's assume that the distance between the charges is d. The force between charges A and B will be given by F_AB = k * q^2 / d^2, where k is the electrostatic constant (9 × 10^9 N m^2/C^2). The force between charges A and C will also be F_AC = k * q^2 / d^2.
7. Resolve the forces mathematically: We know that the forces from B and C cancel each other out in the horizontal direction. Therefore, the net force in the horizontal direction will be zero, resulting in no equation regarding the horizontal force.
8. Apply vertical force equation: To have a net force in the vertical direction, we need to set up the equation F_AB - F_AC = 0, as the forces must add up to zero.
9. Solve for the charge: Substitute the values: F_AB = F_AC, which gives us k * q^2 / d^2 - k * q^2 / d^2 = 0.
10. Simplify the equation: The k, q^2, and d^2 terms cancel out, leaving us with 0 = 0. This suggests that the equation is true for any value of q.
Conclusion: Based on our analysis, we find that any charge magnitude and algebraic sign can be placed at the empty corner to make the net force at corner A point along the vertical direction.