You have three identical charged spheres, each with a charge of 2.80 µC. You have arranged them so they are each at the corners of a square that measures 4m on a side. Find the magnitude of the electric field at the "empty" corner. (The corner with no charged sphere.)

To find the magnitude of the electric field at the "empty" corner, you can use the principle of superposition. The electric field created by each charged sphere can be calculated using Coulomb's law, and then the total electric field at the "empty" corner is the vector sum of the electric fields due to each charged sphere.

Coulomb's law states that the electric field created by a point charge is directly proportional to the charge and inversely proportional to the square of the distance. Mathematically, it is given by:

E = k * (q / r^2),

Where E is the electric field, k is the Coulomb's constant (approximately equal to 9 × 10^9 N·m^2/C^2), q is the charge, and r is the distance between the point charge and the location where you want to find the electric field.

In this case, the charge (q) of each sphere is 2.80 µC (which can be written as 2.80 × 10^-6 C), and the distance (r) between each sphere and the "empty" corner is the diagonal of the square, which can be calculated using Pythagoras' theorem as 4√2 m.

Now, let's calculate the electric field created by each charged sphere at the "empty" corner:

E1 = k * (q / r^2),
E2 = k * (q / r^2),
E3 = k * (q / r^2),

Since the spheres have identical charges and are equidistant to the "empty" corner, the magnitudes of their electric fields will be the same (E1 = E2 = E3). You can then calculate the total electric field at the "empty" corner by adding the vectors:

E_total = E1 + E2 + E3.

Substituting the values we have:

E_total = E + E + E,
E_total = 3E,

where E is the electric field due to one charged sphere.

Now, calculate the value of E using Coulomb's law:

E = k * (q / r^2),
E = (9 × 10^9 N·m^2/C^2) * (2.8 × 10^-6 C / (4√2 m)^2).

Calculating this expression will give you the value of E, which you can then multiply by 3 to get the total electric field at the "empty" corner.