Calculating the Electric Field at the Centroid

To calculate the electric force on a charge q placed at the centroid O of the equilateral triangle, we first need to find the electric field at that point. We can do this by using Coulomb's law and vector addition.

Electric Field Due to a Single Charge

The electric field due to a single charge q at a distance r from it is given by:

E = kq/r^2

where k is the Coulomb constant (9 x 10^9 Nm^2/C^2).

Electric Field Due to Three Charges

To find the electric field due to three charges q, -q, and q placed at the corners of an equilateral triangle of side 'a', we first need to find the electric field due to each charge at the centroid O.

The distance from each charge to the centroid is a/sqrt(3).

Using Coulomb's law, we can find the electric field due to each charge at the centroid:

E1 = kq/(a/sqrt(3))^2 = 3kq/a^2

E2 = k(-q)/(a/sqrt(3))^2 = -3kq/a^2

E3 = kq/(a/sqrt(3))^2 = 3kq/a^2

Resultant Electric Field at the Centroid

To find the resultant electric field at the centroid, we need to add the electric fields due to each charge vectorially. Since the charges are placed at the corners of an equilateral triangle, the three electric fields have the same magnitude but are directed at angles of 120 degrees to each other.

Using vector addition, we can find the resultant electric field at the centroid:

E = 2E1cos(60) = 2(3kq/a^2)(1/2)(1/2) = 3kq/a^2

The electric force on a charge q placed at the centroid O is given by:

F = qE = q(3kq/a^2) = 3kq^2/a^2

Therefore, the resultant electric force on a charge q placed at the centroid O of an equilateral triangle of side 'a' is 3kq^2/a^2.