The drawing shows an equilateral triangle, each side of which has a length of 2.20 cm. Point charges are fixed to each corner, as shown. The 4.00 µC charge experiences a net force due to the charges qA and qB. This net force points vertically downward and has a magnitude of 464 N. Determine the magnitudes and algebraic signs of the charges qA and qB.
To solve this problem, we can use Coulomb's Law, which states that the electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
The formula for Coulomb's Law is:
F = k * (|q1| * |q2|) / r^2
where F is the force between the charges, k is the Coulomb constant (9.0 x 10^9 N.m^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
In this problem, we need to find the magnitudes and algebraic signs of the charges qA and qB.
Step 1: Find the distance between the charges
Since the given triangle is equilateral, each side has a length of 2.20 cm. The distance between the charges qA and qB is the length of one side of the triangle. So, the distance between the charges is 2.20 cm.
Step 2: Find the force between the 4.00 µC charge and the charges qA and qB
We are given that the force between the 4.00 µC charge and the charges qA and qB is 464 N. Since the net force points vertically downward, it means that the forces from qA and qB must be equal and opposite. Therefore, we can distribute the total force equally between qA and qB, resulting in 232 N for each of them.
Step 3: Apply Coulomb's Law to find the charges
Using Coulomb's Law and the given information, we can set up the equations:
k * (4.00 µC * |qA|) / (2.20 cm)^2 = 232 N
k * (4.00 µC * |qB|) / (2.20 cm)^2 = 232 N
Simplifying these equations, we can divide both sides by k and multiply by (2.20 cm)^2:
4.00 µC * |qA| = 232 N * (2.20 cm)^2
4.00 µC * |qB| = 232 N * (2.20 cm)^2
Now we can solve for |qA| and |qB| by plugging in the values in these equations.
Step 4: Determine the algebraic signs of the charges
To determine the algebraic signs of the charges qA and qB, we need to consider the repulsion or attraction between them. Since the net force on the 4.00 µC charge is vertically downward, it means that qA and qB have opposite charges. If qA is positive, then qB must be negative, or vice versa.
Overall, follow these steps to solve the problem:
1. Find the distance between the charges.
2. Divide the given force equally between qA and qB.
3. Apply Coulomb's Law to find the magnitudes of the charges qA and qB.
4. Determine the algebraic signs of the charges based on the net force direction.
After solving these equations and determining the signs, you will find the magnitudes and algebraic signs of the charges qA and qB needed to achieve a net force of 464 N, pointing vertically downward, on the 4.00 µC charge.