Identical point charges of +1.1 ¦ÌC are fixed to three of the four corners of a square. What is the magnitude q of the negative point charge that must be fixed to the fourth corner, so that the charge at the diagonally opposite corner experiences a net force of zero?

To solve this problem, we need to use the concept of electric force and take into account the principle of superposition.

1. The first step is to understand the electric force between two point charges, given by Coulomb's law:

F = k * |q1 * q2| / r^2

Where:
- F is the electric force between the charges
- k is the electrostatic constant (k = 9 × 10^9 N m^2/C^2)
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges

2. We know that the net force experienced by the charge at the diagonally opposite corner must be zero. This means that the total electric force acting on it due to the other three charges must cancel out.

3. We have three charges with the same magnitude (+1.1 µC) fixed to three corners of the square. Let's call the magnitude of the negative charge at the fourth corner as q (unknown).

4. The electric forces between the diagonally opposite charges will be equal in magnitude but opposite in direction. Let's analyze the forces acting on the diagonally opposite charge due to the other three charges:

- Force due to the top-left charge: F1 = k * |(+1.1 µC)(-1.1 µC)| / d^2

- Force due to the top-right charge: F2 = k * |(+1.1 µC)(-1.1 µC)| / d^2

- Force due to the bottom-left charge: F3 = k * |(+1.1 µC)(-1.1 µC)| / d^2

Note that the distances (d) for each force are equal, as all sides of the square would be of equal length.

5. The net force on the diagonally opposite charge will be zero if the total force in the horizontal direction and the total force in the vertical direction cancel each other out.

- Total force in the horizontal direction: F_horizontal = F1 + F2 (equal magnitudes but opposite directions)

- Total force in the vertical direction: F_vertical = F3 (acting in the opposite direction)

For the net force to be zero, F_horizontal = 0 and F_vertical = 0.

6. Setting F_horizontal = 0, we can write:

F1 + F2 = 0
=> k * |(+1.1 µC)(-1.1 µC)| / d^2 + k * |(+1.1 µC)(-1.1 µC)| / d^2 = 0

7. Setting F_vertical = 0, we can write:

F3 = 0
=> k * |(+1.1 µC)(-1.1 µC)| / d^2 = 0

8. Simplifying equations (6) and (7), we get:

2 * k * |(+1.1 µC)(-1.1 µC)| / d^2 = 0

k * |(+1.1 µC)(-1.1 µC)| / d^2 = 0

9. We can see that setting both F_horizontal and F_vertical to zero results in the same equation. This implies that the magnitude of the negative charge (q) at the fourth corner must be zero for the net force on the diagonally opposite charge to be zero.

Therefore, the magnitude (q) of the negative charge that must be fixed to the fourth corner is zero.

Let's consider the charges at each corner of the square:

- Three corners have identical positive charges of +1.1 μC each.
- The charge at the fourth corner is negative.

We need to find the magnitude of the negative charge (q) at the fourth corner, so that the charge at the diagonally opposite corner experiences a net force of zero.

To solve this problem, we can use Coulomb's Law, which states that the electric force between two point charges is given by:

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

Where:
- F is the force between the charges,
- k is Coulomb's constant (k = 8.99 x 10^9 N m^2/C^2),
- q1 and q2 are the magnitudes of the charges,
- r is the distance between the charges.

In this case, we have three charges of +1.1 μC each at three adjacent corners of the square. To find the net force on the charge at the diagonally opposite corner, we need to consider the forces exerted by each of the three positive charges.

Since the charges at the diagonal corners are equidistant, the net force on the diagonally opposite charge will be zero if the magnitudes of the forces due to three positive charges are equal in magnitude and opposite in direction.

Let's consider two adjacent positive charges:

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

The magnitude of the force between these two charges is the same as the magnitude of the net force on the negative charge from both adjacent positive charges.

Now, let's calculate the force between two charges:

F = k * (|q * q| / r^2)

F = k * (|1.1 μC * 1.1 μC| / r^2)

F = k * (1.1^2 × 10^-6 C^2 / r^2)

Since the forces on the diagonally opposite charge are equal in magnitude and opposite in direction, the net force acting on it will be zero.

Therefore, the net force acting on the diagonally opposite charge due to the two adjacent positive charges is:

F_net = F - F (opposite in direction)

Now, let's substitute the values:

0 = k * (1.1^2 × 10^-6 C^2 / r^2) - k * (1.1^2 × 10^-6 C^2 / r^2)

0 = k * (1.1^2 × 10^-6 C^2 / r^2 - 1.1^2 × 10^-6 C^2 / r^2)

0 = k * (0)

Since the forces are equal in magnitude and opposite in direction, their sum will be zero. Therefore, the magnitude of the negative charge at the fourth corner does not affect the net force on the charge at the diagonally opposite corner.

In conclusion, the magnitude of the negative charge (q) that must be fixed to the fourth corner, so that the charge at the diagonally opposite corner experiences a net force of zero, is not affected by the magnitude of the negative charge.

call side of square = 1

look at components along diagonal

2(1.1 *1.1 /1^2)cos 45 = 1.1 q/(sqrt2)^2

cos 45 = 1/sqrt 2

2(1.1) = q/sqrt 2

q = 2.2 sqrt 2