Four identical point charges of q = 4.64 nC are at the four corners of a square with a side length of 13.5 cm as shown in the figure. What charge Q should be placed at the center of the square to keep the other four charges in equilibrum?

Figure the force on one of the corners (the sum of the forces from the three other corners). Then figure Q such that it balances this force.

Force on one corner:

kQq(1/s^2 *.707 + 1/s^2 (.707)+ 1/*sqrt2*s)^2)
= kqq/s^2*(1.414+.5) check that I did it in my head.

that is the force on each corner.

Now, balance that with Q at distance s*.707

forcebalance=kqQ(1/.5s^2)=2kqQ/s^2

solve for Q. Check my work.

so would it be

(2)(9.0e9)(4.64)(Q)/0.707^2

To keep the other four charges in equilibrium, the net electrostatic force on each charge must be zero.

Let's assume that the charge Q at the center of the square is Q = q, where q = 4.64 nC.

At each corner of the square, the electrostatic force on one charge due to the other three charges can be calculated using Coulomb's Law:

F = k * (q1 * q2) / r^2

where F is the electrostatic force, k is the Coulomb's constant (k = 8.99 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.

Since all four charges at the corners are identical and equidistant from the center of the square, the net force on any corner charge will be equal in magnitude and opposite in direction.

Let's consider one of the corner charges:

The distances between the charge at the center and the corner charges can be calculated using the Pythagorean theorem:

d = √(s^2 + s^2) = √(2s^2) = s√2

where d is the distance, and s is the side length of the square.

Now, let's calculate the electrostatic force on one of the corner charges:

F = (8.99 x 10^9 N m^2/C^2) * (q * q) / (s√2)^2

F = (8.99 x 10^9 N m^2/C^2) * (4.64 x 10^-9 C)^2 / (13.5 x 10^-2 m * √2)^2

F = (8.99 x 10^9 N m^2/C^2) * (21.5296 x 10^-18 C^2) / (1.892 x 10^-1 m^2)

F = 8.99 x 21.5296 / 1.892 N ≈ 102.056 N

Since there are four corner charges, the net force on the charge at the center must be the sum of all forces acting on it:

Net force = 4 * 102.056 N = 408.225 N

To keep the net force on the charge at the center zero, a charge Q of -408.225 C should be placed at the center of the square.

Therefore, the charge Q that should be placed at the center of the square to keep the other four charges in equilibrium is approximately -408.225 C.

To find out what charge Q should be placed at the center of the square to keep the other four charges in equilibrium, we need to use the principle of electrostatic equilibrium. According to this principle, in order for an object to be in equilibrium, the net force acting on it must be zero.

In this case, to find the charge Q, we need to calculate the net force acting on it and set it equal to zero.

Let's break down the steps to solve this problem:

Step 1: Calculate the electric force between each of the charges at the corners and Q.
- The electric force between two point charges can be calculated using Coulomb's Law: F = k * (|q1| * |q2|) / r^2
- In this case, the charges at the corners are identical, so we can simplify the equation to: F = k * (q^2) / r^2

Step 2: Determine the direction of the forces.
- Since all the charges q at the corners are the same, they will be repelled by the charge Q at the center. Therefore, the net force on Q will be in the outward direction from the center of the square.

Step 3: Calculate the net force on the charge Q.
- Since the charges at the corners are arranged symmetrically, the net force on Q will be the vector sum of the individual forces acting on it.
- Since the forces are repulsive, the net force acting on Q can be found by summing the magnitudes of the individual forces.

Step 4: Set the net force equal to zero and solve for the charge Q.
- Since the net force acting on Q is in equilibrium, the magnitude of the net force must be zero.
- Set the sum of the magnitudes of the individual forces equal to zero and solve for Q.

Following these steps, we can find the charge Q that should be placed at the center of the square to keep the other four charges in equilibrium.