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?
(the rectangle has +3 uC charges in 3 corners and one empty corner. The long sides are 4d and the short sides are d. A is the bottom left corner. The empty corner is the top right corner)
-3.3microC
To make the net force on the charge at corner A point along the vertical direction, we need to evaluate the electric forces acting on it.
Given that the long sides of the rectangle are 4d and the short sides are d, we can assume that the distances between the charges are as follows:
- The distance between corner A and the charge directly above it is 4d.
- The distance between corner A and the charge directly to the right of it is 2d.
- The distance between corner A and the charge diagonally across from it is 2√5d.
Now, let's evaluate the electric forces using Coulomb's Law:
1. The force exerted on the charge at corner A by the charge directly above it:
F1 = (k * q1 * qA) / r1^2
= (9 * 10^9 N * m^2/C^2) * (3 * 10^-6 C) * (qA) / (4d)^2
2. The force exerted on the charge at corner A by the charge directly to the right of it:
F2 = (k * q2 * qA) / r2^2
= (9 * 10^9 N * m^2/C^2) * (3 * 10^-6 C) * (qA) / (2d)^2
3. The force exerted on the charge at corner A by the charge diagonally across from it:
F3 = (k * q3 * qA) / r3^2
= (9 * 10^9 N * m^2/C^2) * (3 * 10^-6 C) * (qA) / (2√5d)^2
To make the net force along the vertical direction, the vertical components of F1, F2, and F3 must add up to zero.
Considering the signs of the charges involved (as positive or negative), set up the equation:
F1 cos(theta1) + F2 cos(theta2) + F3 cos(theta3) = 0
Since all the angles (theta1, theta2, theta3) are vertical, the cos value for each will be 1.
Plugging in the expressions for F1, F2, and F3, we get:
[(9 * 10^9 N * m^2/C^2) * (3 * 10^-6 C) * (qA) / (4d)^2] + [(9 * 10^9 N * m^2/C^2) * (3 * 10^-6 C) * (qA) / (2d)^2] + [(9 * 10^9 N * m^2/C^2) * (3 * 10^-6 C) * (qA) / (2√5d)^2] = 0
Simplifying the equation, we can solve for qA.
Note: To find the magnitude of qA, disregard the algebraic sign.
Please provide the value of 'd' in order to calculate the magnitude of the charge needed at the empty corner.
To determine the charge and its algebraic sign required to make the net force on the charge at corner A point along the vertical direction, we can use the principle of superposition and apply Coulomb's Law.
Coulomb's Law states that the force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
Here's how we can solve the problem step by step:
1. Assign variables to the distances and charges given in the problem to make it easier to work with. Let's assign the following variables:
- q1: charge at corner A, unknown magnitude and sign
- q2: charge at one of the other corners (+3 uC)
- q3: charge at another corner (+3 uC)
- q4: charge at the empty corner, magnitude and sign to be determined
- d: distance between adjacent corners
- 4d: distance between diagonal corners
2. Since we want the net force on the charge at corner A to be vertical, we need to find the net vertical force acting on it. This means we need to calculate the vertical components of the forces between A and each of the other charges.
3. Calculate the vertical force between A and charge q2:
- The distance between A and q2 is d (one of the short sides).
- According to Coulomb's Law, the force (F2) between q2 and A is given by:
F2 = k * |q1| * |q2| / d^2
- Since we want the force to be vertical, F2 must act upward. This means the magnitude of the force should be |F2| = k * |q1| * |q2| / d^2.
4. Calculate the vertical force between A and charge q3:
- The distance between A and q3 is 4d (one of the long sides).
- Similar to step 3, the vertical force (F3) between q3 and A is given by:
F3 = k * |q1| * |q3| / (4d)^2
- To have a vertical net force, F3 must also act upward. Therefore, the magnitude of the force should be |F3| = k * |q1| * |q3| / (4d)^2.
5. Determine the net vertical force:
- Since F2 and F3 have the same direction (both upward), their magnitudes should be added to obtain the net force (F_net). Both F2 and F3 act in the same direction because the charges at corners q2 and q3 have the same sign.
- F_net = |F2| + |F3|
6. Set up the condition for a net vertical force:
- For the net force to be vertical, F_net = 0. The vertical forces F2 and F3 must cancel each other out.
- Therefore, |F2| + |F3| = 0.
7. Substitute the expressions for F2 and F3 into the condition from step 6 and solve for |q1|:
- k * |q1| * |q2| / d^2 + k * |q1| * |q3| / (4d)^2 = 0
- Simplify the equation by dividing through by |q1|.
- |q2| / d^2 + |q3| / (4d)^2 = 0
- Solve for |q1|. The magnitude of the charge at corner A can be expressed as:
|q1| = - |q2| / d^2 * (4d)^2 / |q3|
8. Calculate the magnitude of the charge at the empty corner, |q4|:
- Since the net force on q1 is vertical, the magnitude of the force (F4) between q1 and q4 should be equal to |F2| + |F3|.
- F4 = k * |q1| * |q4| / (4d)^2
- Substitute the expression for |q1| from step 7 into the equation for F4 to get:
F4 = k * (- |q2| / d^2 * (4d)^2 / |q3|) * |q4| / (4d)^2
- Cancel out common factors to simplify the equation:
F4 = - k * |q2| * |q4| / |q3|
9. Finally, determine the sign of q4 by considering the forces involved:
- Since the forces F2 and F3 act upward, the charge at corner q4 must have a negative sign to create a downward force (F4) that will balance out the upward forces.
- Therefore, the charge at the empty corner, q4, should have a negative sign (-) in front of its magnitude.
To summarize, the charge (magnitude and algebraic sign) that must be placed at the empty corner is |-3 uC|. The charge should have a negative sign (-) to create a downward force that balances out the upward forces.