3 identical spheres are given equal amount of charge and hang from a common point with 3 massless string .each of length l .as that the lines of these charges will form an equilateral triangle .find and the side
To find the side length of the equilateral triangle formed by the charged spheres, we can use the concept of electrostatic equilibrium. The charges on the spheres will distribute themselves in such a way that the net force acting on each sphere is zero.
In this scenario, we have three identical spheres with equal charges, arranged in an equilateral triangle formation. Let's denote the charge on each sphere as Q. Since the charges are the same and the spheres are identical, the electrostatic repulsive forces among the charges will also be the same.
Now, let's consider the forces acting on one of the spheres. We have two other spheres on either side. Due to the symmetry of the equilateral triangle, the charges on these two spheres will also exert forces on the central sphere along the lines joining them.
Since the forces act along the lines joining the spheres, we can resolve these forces into two components: one along the line joining the spheres, and the other perpendicular to this line.
Now, if we take the forces along the lines joining the spheres, we can see that they cancel each other out because the triangle is equilateral. Hence, the net force along the line joining the spheres is zero.
The perpendicular components of the forces will add up to maintain equilibrium. Since the triangle is equilateral, the perpendicular components from the two adjacent spheres will be equal in magnitude and opposite in direction. Therefore, they will cancel each other out.
Hence, we can conclude that there will be no net force acting on each of the spheres, and the charges will be in electrostatic equilibrium.
To find the side length of the equilateral triangle, we can use Coulomb's law, which states that the electrostatic force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
Let's consider one of the spheres. The net force acting on it due to the charges from the other two spheres will be zero (as explained above). Since the forces act along the lines joining the spheres, the magnitude of the force between the charges on the adjacent spheres will be given by Coulomb's law.
The force of repulsion between two equal charges Q at a distance L is given by:
F = k * Q^2 / L^2
where k is the electrostatic constant.
In the equilateral triangle, the distance between the centers of adjacent spheres is equal to the side length of the triangle (which we need to find). Let's denote it by 'a'.
Therefore, using Coulomb's law, we can write:
k * Q^2 / a^2 = k * Q^2 / L^2
Simplifying the equation:
a^2 = L^2
Taking the square root of both sides:
a = L
Hence, the side length of the equilateral triangle formed by the charges is equal to the length of the string, which is given as 'L' in the problem statement.
So, the side length (a) of the equilateral triangle is equal to L.