Equilibrium of Four Point Charges in a Square: Calculating Charge Q

[badd99]

Four identical point charges of q = 8.37 nC are at the four corners of a square with a side length of 11.2 cm as shown in the figure. (had to make it but same thing since I can't copy it)

q q

Q

q q

Edit: doesn't come out right when I post so imagen a square little q's on corners and big Q in the center.

What charge Q should be placed at the center of the square to keep the other four charges in equilibrium?

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I have no idea how to even start. There are so many vectors here I can't figure out where to even beginning making things equal electrical pull/push

 

[Tomer]

Well, start by specifically writing the forces acting by 3 charges on the fourth one. You should know similar signs reject each other and different signs attract. By calculating the sum of the forces, see what force it is that will balance it.

The same charge will work for all other charges because of the symmetry of the problem.

 

[Doc Al]

Start by picking one of the four charges (the upper right, for instance) and figure out the force that each of the other three exert on it. Once you find the net force from those three, you can figure out what Q would have to be to counter that force.

(Edit: Looks like Tomer beat me to it.)

 

[Spinnor]

You should know you need to put a negative charge at the center to balance the forces. Look at the forces on any corner charge. There will be four forces that act on it, the three forces from the other corner charges and the force from the charge at the middle.

Rotate the square so that you only need to consider forces in the x direction. See the attached sketch. Hope that helps.

 

[rwisz]

. I understand your confusion and I am here to help. The concept of equilibrium in electrostatics is based on the principle that the net force on any charge must be zero. In this case, we have four charges at the corners of a square and we need to find the charge at the center that will keep them in equilibrium.

To solve this problem, we can use the concept of electric field and its vector nature. The electric field is a vector quantity that describes the strength and direction of the electric force at a particular point in space. In this case, we have four electric fields created by the four charges at the corners of the square.

To find the electric field at the center of the square, we can use the principle of superposition, which states that the net electric field at a point due to multiple charges is the vector sum of the individual electric fields. In other words, we can add the electric fields created by each charge at the center to find the total electric field.

Now, since we want the net force on the center charge to be zero, we need to find the charge Q that will produce an electric field that will exactly cancel out the electric fields created by the four corner charges.

To do this, we can use the equation for electric field:

E = k * q / r^2

Where k is the Coulomb's constant, q is the charge creating the electric field, and r is the distance between the charge and the point where we want to find the electric field.

In this case, the distance between each corner charge and the center is the same, 11.2 cm. So, we can write the equation for the net electric field at the center as:

E_net = E1 + E2 + E3 + E4 = k * q / (11.2 cm)^2 + k * q / (11.2 cm)^2 + k * q / (11.2 cm)^2 + k * q / (11.2 cm)^2

Since the charges at the corners are all identical, we can simplify this equation to:

E_net = 4 * k * q / (11.2 cm)^2

Now, we want this net electric field to be zero, so we can set it equal to zero and solve for Q:

0 = 4 * k * q / (11.2 cm)^2

Q = -4 * k * q / (