To solve this problem, we need to analyze the forces acting on the negatively charged particle 'P' due to the charged strip and then apply the principles of energy conservation to find the velocity of 'P' as it moves to the specified point. Let's break this down step by step.

Understanding the Setup

We have a strip of length 'l' with a uniform linear charge density '$'. This means that the total charge on the strip can be expressed as:

• Total charge, Q = $ * l

The negatively charged particle 'P' has a charge of '-q' and is initially at a distance 'd' from the end 'A' of the strip. We want to find its velocity when it reaches a point that is at a distance 'd/2' from 'A'.

Electric Field Due to the Charged Strip

The electric field (E) created by a uniformly charged infinite line of charge at a distance 'r' from it is given by the formula:

• E = (1 / (4πε₀)) * (2Q / r)

However, since our strip is finite, we need to consider the contributions from both ends of the strip. For simplicity, we can approximate the electric field at point 'P' due to the strip as:

• E ≈ (1 / (4πε₀)) * (2Q / d)

This approximation assumes that 'd' is much larger than the length of the strip, which is often valid in practical scenarios.

Force on the Particle

The force (F) acting on the particle 'P' due to the electric field is given by:

• F = -qE

Substituting the expression for E, we get:

• F = -q * (1 / (4πε₀)) * (2Q / d)

This force will act in the direction of the electric field, which is towards the strip since 'P' is negatively charged.

Work Done and Energy Conservation

As the particle moves from distance 'd' to 'd/2', it experiences a change in potential energy. The work done (W) on the particle as it moves through this distance can be calculated by integrating the force over the distance:

• W = ∫(F) dx from d to d/2

However, a more straightforward approach is to use the conservation of energy. The initial potential energy (U_initial) when the particle is at distance 'd' can be expressed as:

• U_initial = -q * V(d)

And the potential energy when it reaches distance 'd/2' is:

• U_final = -q * V(d/2)

Using the relationship between potential and electric field, we can express the change in potential energy as:

• ΔU = U_final - U_initial = -q * (V(d/2) - V(d))

Finding the Velocity

According to the work-energy theorem, the work done on the particle is equal to its change in kinetic energy:

• W = ΔK.E = (1/2)mv^2 - 0

Setting the work done equal to the change in kinetic energy gives us:

• ΔU = (1/2)mv^2

From this equation, we can solve for the velocity 'v' of the particle 'P' when it reaches the point at distance 'd/2'. Rearranging gives:

• v = sqrt((2 * ΔU) / m)

Final Expression

By substituting the expressions for potential energy and electric field into this equation, we can derive the final expression for the velocity of 'P'. The exact calculations will depend on the specific values of '$', 'l', 'q', 'm', and 'd'. However, the key takeaway is that the velocity can be determined using the principles of electric fields, forces, and energy conservation.

In summary, by understanding the forces acting on the charged particle and applying the conservation of energy, we can effectively find the velocity of 'P' as it moves towards the charged strip.