Four charged particles are arranged at the corners (1, 2, 3 and 4) of a square with each side length L The particles at opposite corners have equal charge.
a). What will be the sign of charges at each corner to make net
electric force on each charge at corner 1 and 4 equal to 0.
b). Find the relationship between Q and q such that the electric
force on each particle with charge Q is zero.
To find the sign of charges at each corner to make the net electric force on charges at corner 1 and 4 equal to zero, we first need to understand the concept of electric forces.
The electric force between two charged particles is given by Coulomb's Law:
F = k * (q1 * q2) / r^2
Where:
F is the magnitude of the electric force
k is the electrostatic constant (k ≈ 9 x 10^9 Nm^2/C^2)
q1 and q2 are the charges of the particles
r is the distance between the particles
a) To make the net electric force on charges at corner 1 and 4 equal to zero, we need to ensure that the forces acting on these charges from the other charges balance out.
Let's start by analyzing the forces on charge at corner 1:
- From charge at corner 2: The distance between the charges is L, and the force is given by F12 = k * (q1 * q2) / L^2.
- From charge at corner 3: The distance between the charges is L√2, and the force is given by F13 = k * (q1 * q3) / (L√2)^2.
- From charge at corner 4: The distance between the charges is L√2, and the force is given by F14 = k * (q1 * q4) / (L√2)^2.
Since the net electric force on charge at corner 1 is zero, we have:
F12 + F13 + F14 = 0
Now, let's analyze the forces on charge at corner 4:
- From charge at corner 1: The distance between the charges is L√2, and the force is given by F41 = k * (q4 * q1) / (L√2)^2.
- From charge at corner 2: The distance between the charges is L, and the force is given by F42 = k * (q4 * q2) / L^2.
- From charge at corner 3: The distance between the charges is L√2, and the force is given by F43 = k * (q4 * q3) / (L√2)^2.
Again, since the net electric force on charge at corner 4 is zero, we have:
F41 + F42 + F43 = 0
Now, we can substitute the respective values of F12, F13, F14, F41, F42, and F43 and solve for the unknowns q1, q2, q3, and q4 to make the equations hold.
b) To find the relationship between Q and q such that the electric force on each particle with charge Q is zero, we can use a similar approach.
Let's consider the force acting on charge at any corner (let's say corner 1) due to charge Q:
- From charge at corner 2: The distance between the charges is L, and the force is given by F12 = k * (Q * q2) / L^2.
- From charge at corner 3: The distance between the charges is L√2, and the force is given by F13 = k * (Q * q3) / (L√2)^2.
- From charge at corner 4: The distance between the charges is L√2, and the force is given by F14 = k * (Q * q4) / (L√2)^2.
Since the net electric force on charge at corner 1 is zero, we have:
F12 + F13 + F14 = 0
Similarly, we can analyze the forces acting on charges at corners 2, 3, and 4 to derive similar equations.
By solving these equations, we can find the relationship between Q and q to make the electric forces on each particle with charge Q equal to zero. The specific values of charges at corners 2, 3, and 4 will depend on the distances and the signs of the charges at corners 1, 2, 3, and 4.