Three charges -q q-q are placed at the corner of an equilateral triangle of side a the resultant electric force on a charge q placed at it's centroid o of the triangle?

Question

Three charges -q  q-q are placed at the corner of an equilateral triangle of side a the resultant electric force on a charge  q placed at it's centroid o of the triangle?

Answer

Electric force on a charge placed at centroid of an equilateral triangle

Given

Three charges -q, q, -q are placed at the corners of an equilateral triangle of side a

A charge q is placed at the centroid O of the triangle

Explanation

The electric force acting on a charge q placed at the centroid O of the triangle can be found by applying Coulomb's Law

Let F1, F2, and F3 be the electric forces acting on the charge q due to the charges -q, q, and -q respectively

Let r be the distance between the charge q and the charges -q, q, and -q respectively

By symmetry, the forces F1 and F3 will cancel each other out

Therefore, the resultant force on the charge q will be the vector sum of F2 and the negative of F1 or F3

The magnitude of the electric force can be found using Coulomb's Law:

F = k * q1 * q2 / r^2

where k is the Coulomb's constant, q1 and q2 are the magnitudes of the charges, and r is the distance between the charges

The direction of the force can be found using the principle of superposition, which states that the net force is the vector sum of all the individual forces

Using the distance formula, the distance between the charge q and the charges -q, q, and -q respectively can be found to be:

r = a / sqrt(3)

Substituting the values of the charges and the distance into Coulomb's Law, the magnitudes of F1, F2, and F3 can be found to be:

F1 = F3 = k * q * q / (a / sqrt(3))^2 = (3 * k * q^2) / a^2

F2 = k * q * (-q) / (2a / sqrt(3))^2 = (-4 * k * q^2) / (3a^2)

Therefore, the net force on the charge q is:

Fnet = F2 - F1 = (-7 * k * q^2) / (3a^2)

Conclusion

The resultant electric force acting on a charge q placed at the centroid O of an equilateral triangle of side a and with charges -q, q, and -q placed at its corners is (-7 * k * q^2) / (3a^2)

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