Identical point charges of +2.1 ìC are fixed to diagonally opposite corners of a square. A third charge is then fixed at the center of the square, such that it causes the potentials at the empty corners to change signs without changing magnitudes. Find the sign and magnitude of the third charge.

Well, isn't this a shocking situation in a square! Let's dive right into the electrifying details.

We have two identical point charges of +2.1 μC, or micro-Coulombs, fixed to diagonally opposite corners of a square. Now, we know that charges of the same sign repel each other, so there must be some sparks flying here.

To counteract this electrifying situation, we need to introduce a third charge at the center of the square. This charge should cause the potentials at the empty corners to change signs without changing magnitudes. Oh, the suspense!

Since the charges at the corners have the same magnitudes, but we want the potentials to change sign, we need to add a charge of the opposite sign at the center of the square. It's a positive-negative dance!

To figure out the magnitude of this charge, let's consider that the potential at the empty corners changes signs but not magnitudes. This means that the magnitude of the new charge at the center should be the same as the magnitudes at the corners, which is +2.1 μC. So, the magnitude of the third charge is 2.1 μC.

Now, as for the sign, we already know that the new charge should be of the opposite sign to generate a potential change. Since the original charges at the corners are positive, the third charge should be negative. That's right, it's time to bring in some negativity!

Therefore, the sign of the third charge is negative, and its magnitude is 2.1 μC. Negative charge to the rescue, saving the day in this electrifying square dance!

To solve this problem, we need to consider the electrostatic potential at the empty corners of the square.

Let's assume that the distance between the charges at the diagonal corners is "d". The potential at the empty corners can be calculated using the formula for electric potential due to a point charge Q at a distance r:

V = k * Q / r,

where V is the potential, k is the Coulomb's constant, Q is the charge, and r is the distance from the charge.

Initially, the potential at the empty corners is positive since the charges at the diagonal corners are positive (+2.1 μC).

To find the sign and magnitude of the third charge at the center of the square, we need to make the potential negative without changing its magnitude.

Let's assume the magnitude of the third charge at the center is Q3. Since the diagonal charges are identical, the distance between the center charge and each diagonal charge is d/√2.

The potential at the empty corners is given by the sum of the potentials due to the diagonal charges minus the potential due to the center charge.

Initially, the potential at the empty corners is:

V_initial = (k * Q) / (d/√2) + (k * Q) / (d/√2).

To make the potential negative, we add the potential due to the third charge at the center, Q3:

V_final = (k * Q) / (d/√2) + (k * Q) / (d/√2) - (k * Q3) / (d/√2) = - (k * Q3) / (d/√2).

To make the magnitudes of V_initial and V_final the same, we set them equal:

(k * Q) / (d/√2) + (k * Q) / (d/√2) = - (k * Q3) / (d/√2).

Simplifying, we get:

2 * (k * Q) / (d/√2) = - (k * Q3) / (d/√2).

Canceling common terms, we have:

2 * Q = - Q3.

Therefore, the magnitude of the third charge at the center of the square is 2 times the magnitude of the charges at the diagonal corners, and the sign is negative.

Hence, the magnitude of the third charge is 2 * 2.1 μC = 4.2 μC, and its sign is negative (-).

To solve this problem, we can use the concept of electric potential. The electric potential at a point in space due to a point charge is given by the equation:

V = k * q / r

Where:
- V is the electric potential
- k is the electrostatic constant (9 x 10^9 Nm^2/C^2)
- q is the charge of the point charge
- r is the distance between the point charge and the point where the electric potential is being calculated.

In this problem, we have two point charges fixed at diagonally opposite corners of a square, and we need to determine the sign and magnitude of the third charge placed at the center of the square.

Let's assume the charge at the center of the square is q3. The potentials at the empty corners of the square change signs when the third charge is added. This means that the potential at one corner changes from positive to negative, while the potential at the other corner changes from negative to positive.

The electric potential at a point due to multiple point charges is simply the sum of the contributions from each individual charge. Since the charges at the diagonal corners are identical, their contributions cancel each other out at the empty corners.

Therefore, the magnitude of the electric potential at the empty corners of the square due to the two corner charges is zero before the third charge is added.

When the third charge q3 is added at the center, it produces a potential at the empty corners. To retain the same magnitude of potential (zero) at the empty corners while changing sign, we must have:

Potential due to q3 = - Potential due to corner charges

Mathematically, this can be written as:

k * q3 / L = - k * q / D

Where:
- L is the length of the square (diagonal distance between the corners)
- D is the distance between the charge at the corner and the center of the square.

Since the charges at the corners are identical (+2.1 µC) and the square is symmetric, we have:

D = L / √2

Now we can solve the equation for q3:

q3 = - q * (L / D)
= - q * (L / (L / √2))
= - q * √2

Plugging in the values, we have:

q3 = - (+2.1 µC) * √2

Therefore, the sign of the third charge is negative, and the magnitude is approximately equal to 2.97 µC.

q=2.1μC=2.1•10⁻⁶ C,

the side of square ia “a” .
The potential at the empty corner due to two charges
φ=φ₁+φ₂ = kq/a + kq/a= 2kq/a
The potential at the empty corner due to three charges
φ`=φ₁+φ₂ +φ₃= kq/a + kq/a + kq₀2/(a√2)= 2kq/a + kq₀2/(a√2).
By the data
φ`= - 2kq/a, =>
2kq/a + kq₀2/(a√2) = - 2kq/a
kq₀2/(a√2) = - 4kq/a
q₀=2√2q=2•1.41•2.1•10⁻⁶ =5.94•10⁻⁶ C