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 charge must be of the same magnitude as the charge at corner A, but with the opposite algebraic sign.
To determine the charge magnitude and algebraic sign that must be placed at the empty corner of the rectangle in order to make the net force on the charge at corner A point along the vertical direction, we can follow these steps:
Step 1: Analyze the forces acting on the charge at corner A:
- There are two forces acting on the charge at corner A due to the nearby charges at corners B and C.
- Let's label the charge at corner B as q₁ and the charge at corner C as q₂.
Step 2: Determine the direction of the forces:
- Due to the principles of electromagnetism, like charges repel each other, and opposite charges attract each other.
- Since the net force on the charge at corner A needs to point along the vertical direction, the forces from charges q₁ and q₂ must cancel each other out horizontally, while pointing upwards vertically.
Step 3: Calculate the electric forces between the charges:
- The electric force between two charges q and Q separated by a distance r is given by Coulomb's Law: F = k * (|q| * |Q|) / r², where k is the electrostatic constant.
- In this case, we can assume that the distances between the charges are equal and define it as "d".
Step 4: Set up the equation:
- Since we need the two forces to cancel each other out horizontally, |F₁| = |F₂|.
- Setting up the equation using Coulomb's Law for both forces, we have: k * (|q₁| * |qA|) / d² = k * (|q₂| * |qA|) / d².
Step 5: Simplify and solve for |qA|:
- We can cancel out the constants k and d² from both sides of the equation.
- The equation simplifies to: |q₁| = |q₂|.
- This means that the magnitude of the charge at corner A must be equal to the magnitude of the charges at corners B and C.
Step 6: Determine the algebraic sign:
- Since we know that the forces from q₁ and q₂ need to cancel each other out horizontally, the algebraic sign of the charge at corner A must be opposite to that of the charges at corners B and C.
- If q₁ and q₂ have opposite signs, the charge at corner A must have the same sign as q₁ (positive or negative, depending on the algebraic sign of q₁).
In conclusion, the charge at the empty corner should have the same magnitude as the charges at corners B and C, but with the opposite algebraic sign.
To solve this problem, we need to analyze the forces acting on the charge at corner A.
First, let's label the corners of the rectangle as A, B, C, and D.
Next, we need to consider the forces from the charges at the other three corners. Let's assume that the charges at B, C, and D are q1, q2, and q3, respectively.
The force between two charges is given by Coulomb's Law:
F = k * (|q1| * |q2|) / r^2,
where F is the force, k is Coulomb's constant (approximately 9 × 10^9 N m^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.
Now, let's consider the forces acting on the charge at corner A:
1. The force from the charge at corner B: This force is directed towards the left side of the rectangle. We'll represent it as F_AB.
2. The force from the charge at corner C: This force is directed towards the right side of the rectangle. We'll represent it as F_AC.
3. The force from the charge at corner D: This force is directed upward. We'll represent it as F_AD.
Since we want the net force on the charge at corner A to point vertically, the horizontal forces F_AB and F_AC should cancel each other out. Their magnitudes should be equal.
So, to determine the magnitude of the charge q1 at corner B, we can equate the forces F_AB and F_AC:
k * (|q| * |q1|) / r^2 = k * (|q| * |q2|) / r^2,
where |q| represents the magnitude of the charge at corner A.
Since the distances between the charges are the same, the magnitudes of the charges must be equal, resulting in |q1| = |q2| = |q|.
Now, let's consider the force from the charge at corner D, F_AD. Since this force should cancel out the horizontal forces, its magnitude should be equal to the sum of the magnitudes of the horizontal forces. This means:
F_AD = F_AB + F_AC,
k * (|q| * |q3|) / r^2 = 2 * (k * (|q| * |q1|) / r^2).
Simplifying this equation, we find:
|q3| = 2 * |q1|.
Therefore, the magnitude of the charge at corner D, |q3|, should be twice the magnitude of the charge at corner B, |q1|.
The sign of the charge depends on the direction of the force. If the force is attractive (pointing towards the center of the rectangle), the charge must have an opposite sign to the charge at corner A. If the force is repulsive (pointing away from the center), the charge at corner D must have the same sign as the charge at corner A.
In conclusion, to make the net force on the charge at corner A point vertically, you need to place a charge with magnitude |q1| at the empty corner (let's call it E). The sign of the charge at corner E should be the same as the charge at corner A.