two identical cahrges particle having charges q and mass m are placed L distance apart. if one charge is fixed and other free to move . find its speed when distance between them becomes to 2L

To solve the problem of two identical charged particles, where one is fixed and the other is free to move, we can apply the principles of electrostatics and conservation of energy. Let's break this down step by step.

Understanding the Forces at Play

When two identical charges, each with charge \( q \), are placed a distance \( L \) apart, they experience a repulsive electrostatic force due to Coulomb's law. The force \( F \) between the charges can be expressed as:

F = k * (q^2) / r^2

where \( k \) is Coulomb's constant, and \( r \) is the distance between the charges. Initially, \( r = L \).

Initial Setup

At the start, the potential energy \( U \) of the system when the charges are at distance \( L \) is given by:

U_initial = k * (q^2) / L

As the free charge moves away to a distance of \( 2L \), the potential energy at that point becomes:

U_final = k * (q^2) / (2L)

Energy Conservation Principle

According to the conservation of mechanical energy, the total mechanical energy of the system remains constant if only conservative forces are acting. Therefore, the initial potential energy will convert into kinetic energy as the charge moves away. The kinetic energy \( K \) of the moving charge when it reaches a distance of \( 2L \) can be expressed as:

K = (1/2) * m * v^2

where \( v \) is the speed of the moving charge. Setting the initial potential energy equal to the sum of the final potential energy and the kinetic energy gives us:

U_initial = U_final + K

Setting Up the Equation

Substituting the expressions for potential energy and kinetic energy into the equation, we have:

k * (q^2) / L = k * (q^2) / (2L) + (1/2) * m * v^2

Solving for Speed

Now, let's simplify this equation. First, we can eliminate \( k * (q^2) \) from both sides:

1 / L = 1 / (2L) + (1/2) * (v^2 / m)

Multiplying through by \( 2L \) to eliminate the denominators gives:

2 = 1 + (L * v^2) / m

Rearranging this leads to:

(L * v^2) / m = 1

From here, we can solve for \( v^2 \):

v^2 = m / L

Taking the square root gives us the final expression for the speed:

v = sqrt(m / L)

Final Thoughts

Thus, the speed of the moving charge when the distance between the two charges becomes \( 2L \) is given by \( v = \sqrt{m / L} \). This result illustrates how energy conservation principles can effectively describe the motion of charged particles under electrostatic forces.