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.)
Apply Coulomb's law for the individual E-fields; then add them as vectors. The resultant will be directed along a diagonal.
To find the magnitude of the electric field at the "empty" corner, we can use the principle of superposition. This principle states that the total electric field at a point due to a collection of charges is the vector sum of the electric fields produced by each individual charge.
Here is how we can calculate the electric field at the "empty" corner:
1. Calculate the electric field produced by each charged sphere individually using the formula for electric field:
Electric field (E) = (k * |Q|) / r^2,
where k is the Coulomb's constant (8.99 x 10^9 N⋅m^2/C^2), |Q| is the magnitude of the charge, and r is the distance between the charge and the point where we are calculating the electric field.
2. Since the spheres are identical and the distances from each sphere to the "empty" corner are the same, the electric field produced by each sphere will have the same magnitude and point in the same direction (towards the "empty" corner).
3. Use the principle of superposition to find the total electric field at the "empty" corner by adding the electric fields produced by each sphere as vectors.
Let's go ahead and calculate the electric field at the "empty" corner.
1. Calculate the electric field produced by each charged sphere:
E1 = (k * |Q|) / r1^2,
E2 = (k * |Q|) / r2^2,
E3 = (k * |Q|) / r3^2,
where |Q| = 2.80 µC = 2.80 x 10^-6 C is the magnitude of the charge, and r1 = r2 = r3 = 4 m is the distance between each charged sphere and the "empty" corner.
Substituting the values, we have:
E1 = E2 = E3 = (8.99 x 10^9 N⋅m^2/C^2 * 2.80 x 10^-6 C) / (4 m)^2.
2. Calculate the electric field magnitude produced by each sphere:
E1 = E2 = E3 = (8.99 x 10^9 N⋅m^2/C^2 * 2.80 x 10^-6 C) / (4 m)^2 = 1.574875 N/C.
3. Use the principle of superposition to find the total electric field:
Since the electric fields produced by each sphere have the same magnitude and point in the same direction towards the "empty" corner, we can add them as vectors:
E_total = E1 + E2 + E3 = 1.574875 N/C + 1.574875 N/C + 1.574875 N/C = 4.72463 N/C.
The magnitude of the electric field at the "empty" corner is approximately 4.72463 N/C.