Two small beads having positive charges 3q and q are fixed at the opposite ends of a horizontal insulating rod extending from the origin to the point x= d. A third small charged bead is free to slide on the rod, at what position is the third bead in equilibrium.

To determine the position of the third charged bead in equilibrium on the insulating rod, we need to analyze the forces acting on it due to the other two charged beads. The beads with charges 3q and q are fixed at the ends of the rod, and the third bead, which we'll denote as having charge Q, can slide freely along the rod. The goal is to find the point where the net force acting on the third bead is zero.

Understanding the Forces

Since the beads have positive charges, they will repel each other. The force exerted on the third bead by the bead with charge 3q will be directed away from it, and similarly, the force from the bead with charge q will also push it away. For the third bead to be in equilibrium, the repulsive forces from both beads must balance each other out.

Setting Up the Problem

Let’s denote the position of the third bead as x, where x is measured from the origin (the position of the bead with charge 3q). The distance from the bead with charge 3q to the third bead is simply x, and the distance from the third bead to the bead with charge q (located at x = d) is (d - x).

Calculating the Forces

The electrostatic force between two point charges can be calculated using Coulomb's law, which states:

• F = k * |q1 * q2| / r²

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

For the third bead (charge Q), the force due to the bead with charge 3q is:

• F₁ = k * |3q * Q| / x²

And the force due to the bead with charge q is:

• F₂ = k * |q * Q| / (d - x)²

Equilibrium Condition

For the third bead to be in equilibrium, these two forces must be equal:

• F₁ = F₂

Substituting the expressions for F₁ and F₂ gives us:

• k * |3q * Q| / x² = k * |q * Q| / (d - x)²

We can simplify this equation by canceling out k and Q (assuming Q is not zero):

• 3q / x² = q / (d - x)²

Next, we can divide both sides by q (again assuming q is not zero):

• 3 / x² = 1 / (d - x)²

Solving for x

Cross-multiplying gives:

• 3(d - x)² = x²

Expanding this results in:

• 3(d² - 2dx + x²) = x²

Which simplifies to:

• 3d² - 6dx + 3x² = x²

Rearranging terms leads to:

• 2x² - 6dx + 3d² = 0

Using the Quadratic Formula

This is a standard quadratic equation of the form ax² + bx + c = 0, where:

• a = 2
• b = -6d
• c = 3d²

We can apply the quadratic formula:

• x = [ -b ± √(b² - 4ac) ] / 2a

Substituting in our values gives:

• x = [ 6d ± √((-6d)² - 4 * 2 * 3d²) ] / (2 * 2)

Calculating the discriminant:

• 36d² - 24d² = 12d²

Thus, we have:

• x = [ 6d ± 2√(3)d ] / 4

This simplifies to:

• x = (3d ± √(3)d) / 2

Finding the Valid Solution

We have two potential solutions for x:

• x₁ = (3 + √3)d / 2
• x₂ = (3 - √3)d / 2

Since x must be between 0 and d, we need to evaluate these solutions. The first solution, x₁, exceeds d, which is not physically possible. Thus, the valid position for the third bead in equilibrium is:

• x = (3 - √3)d / 2

This position ensures that the forces acting on the third bead are balanced, allowing it to remain in equilibrium on the insulating rod.