Three equal charge Q each are place in the vertices of an equilateral triangle of side a.What is the resultant force on any one charge due to the other two?
(2 k Q^2/a^2) cos 30
To find the resultant force on one charge due to the other two charges, we need to calculate the net force acting on that charge.
First, let's label the charges A, B, and C, with A being the charge we want to find the resultant force on, and B and C being the other two charges.
Step 1: Calculate the electric force between charge A and charge B.
The electric force between two charges is given by Coulomb's Law:
F_AB = k * (q_A * q_B) / r_AB^2
where F_AB is the force between charges A and B, k is the electrostatic constant (9 x 10^9 Nm^2/C^2), q_A and q_B are the magnitudes of charges A and B respectively, and r_AB is the distance between charges A and B.
Since the charges are at the vertices of an equilateral triangle, the distance between any two charges is equal to the side length of the triangle, a.
F_AB = k * (Q * Q) / a^2
Step 2: Calculate the electric force between charge A and charge C.
Similarly, the electric force between charges A and C is:
F_AC = k * (q_A * q_C) / r_AC^2
where F_AC is the force between charges A and C, q_A and q_C are the magnitudes of charges A and C respectively, and r_AC is the distance between charges A and C.
Since the charges are at the vertices of an equilateral triangle, the distance between any two charges is equal to the side length of the triangle, a.
F_AC = k * (Q * Q) / a^2
Step 3: Calculate the net force on charge A.
To find the net force on charge A, we need to calculate the vector sum of forces F_AB and F_AC. Since they act in different directions (angle of 60 degrees between them), we can use vector addition:
F_net = √((F_AB)^2 + (F_AC)^2 + 2 * F_AB * F_AC * cosθ)
where θ is the angle between F_AB and F_AC, which is 60 degrees.
F_net = √((2 * k * (Q * Q) / a^2)^2 + 2 * (2 * k * (Q * Q) / a^2) * (2 * k * (Q * Q) / a^2) * cos60)
Simplifying this expression will give you the magnitude of the net force (F_net) on charge A due to the other two charges.
Note: The direction of the net force will depend on the relative positions of the charges in the triangle.