Let $\lambda = q/l$ be the linear charge density. We analyze the work done by the electric forces as the rod enters the fields.

1. Force Profile:

• Phase 1 ($0 < x < d$): The leading end is in Region 1. The retarding force increases linearly as more charge enters the field. $$ F(x) = -(\lambda x) E $$ Work done: $W_1 = \int_0^d -\lambda E x \, dx = -\frac{1}{2} \lambda E d^2$.
• Phase 2 ($d < x < 2d$): The leading end enters Region 2 (Accelerating), while the segment of length $d$ behind it is still in Region 1 (Retarding).
  Force from Region 1: $F_1 = -\lambda d E$ (Constant max retarding force).
  Force from Region 2: $F_2 = +\lambda (x-d) E$ (Increasing accelerating force).
  Net Force: $F_{net} = \lambda E (x – 2d)$.
  This force is still negative (retarding) throughout this interval, reaching zero only at $x = 2d$. Thus, the rod continues to lose energy until $x = 2d$.
  Work done: $W_2 = \int_d^{2d} \lambda E (x – 2d) \, dx = -\frac{1}{2} \lambda E d^2$.

2. Total Energy Barrier:

The total work done against the fields to reach the point $x=2d$ (where net force becomes zero) is:

$$ W_{total} = W_1 + W_2 = -\frac{1}{2} \lambda E d^2 – \frac{1}{2} \lambda E d^2 = -\lambda E d^2 $$

To pass through, the initial kinetic energy must overcome this barrier:

$$ K_i \ge |W_{total}| $$ $$ \frac{1}{2} m u^2 = \lambda E d^2 $$ $$ \frac{1}{2} m u^2 = \left( \frac{q}{l} \right) E d^2 $$

Solving for $u$:

$$ u = \sqrt{\frac{2 q E d^2}{m l}} = d \sqrt{\frac{2 q E}{m l}} $$

The electric fields are symmetric (equal magnitude, opposite direction, equal width). The electric potential $V(x)$ formed by this configuration starts at 0, rises to a peak, and returns to 0 after the second region.

Since the electric force is conservative and the potential at the start (far left) is equal to the potential at the end (far right), the total change in potential energy for the rod traversing the entire system is zero ($\Delta U = 0$).

By conservation of energy, $\Delta K = 0$. Therefore, the exit velocity is equal to the entrance velocity.