three electric charges, 3e, 3e and 12e, where "e" is proton charge, are placed at three of four vertices of a square of side 3cm. Calculate the electric field in the center of the square.

To calculate the electric field at the center of the square, we can break down the problem into two steps.

Step 1: Calculate the electric field component due to each charge separately.
Step 2: Calculate the total electric field at the center by vector addition.

Step 1: Calculate the electric field due to each charge.
The electric field at a point due to a point charge is given by Coulomb's Law:

E = k * (q / r^2)

Where:
- E is the electric field,
- k is the Coulomb's constant (k = 9 × 10^9 N m^2/C^2),
- q is the charge,
- r is the distance between the charge and the point where we want to calculate the field.

For the first charge (3e) at one vertex of the square:
q₁ = 3e
r₁ = diagonal of the square (which can be calculated as √(s^2 + s^2), where s is the side of the square)

For the second charge (3e) at another vertex:
q₂ = 3e
r₂ = diagonal of the square

And for the third charge (12e) at the remaining vertex:
q₃ = 12e
r₃ = diagonal of the square

Step 2: Calculate the total electric field at the center by vector addition.
The electric field obeys the principle of superposition, which means the total electric field at a point due to multiple charges is found by vector addition of the individual electric fields.

E_total = E₁ + E₂ + E₃

Once we have the individual electric fields due to each charge, we can calculate the total electric field at the center of the square by adding up the vector components.

Now let's calculate the values and perform the calculations:

For the diagonal of the square:
r = √(3^2 + 3^2) = √(18) = 3√2 cm

For the first charge (3e):
q₁ = 3e
r₁ = 3√2 cm
E₁ = k * (q₁ / r₁^2) = (9 × 10^9 N m^2/C^2) * (3e / (3√2 cm)^2)

For the second charge (3e):
q₂ = 3e
r₂ = 3√2 cm
E₂ = k * (q₂ / r₂^2) = (9 × 10^9 N m^2/C^2) * (3e / (3√2 cm)^2)

For the third charge (12e):
q₃ = 12e
r₃ = 3√2 cm
E₃ = k * (q₃ / r₃^2) = (9 × 10^9 N m^2/C^2) * (12e / (3√2 cm)^2)

Finally, add up the electric fields to obtain the total electric field at the center of the square:

E_total = E₁ + E₂ + E₃

Please calculate the values for charges q₁, q₂, q₃ and substitute them into the formulas to find E₁, E₂, and E₃. Then add up E₁, E₂, and E₃ to find E_total.