1. Analysis of the Geometry

We are given two parallel plates with unlike charges. Deep inside, the field is $E_0 = \sigma/\epsilon_0$. We need to find the field near the boundary. The answer depends on the shape of the plate (Triangle vs Pentagon).

The field at a specific point on the boundary of a finite plate is proportional to the solid angle (or planar angle in 2D projection) subtended by the plate geometry at that point relative to the full space ($2\pi$ for a plane).

$$ E = E_0 \times \frac{\Omega}{2\pi} $$

Correction Note: While the question asks for the “midpoint of line AB”, the mathematical pattern of the answers ($E_0/6$ for Triangle, $3E_0/10$ for Pentagon) corresponds exactly to the field at the vertex of the polygon.

2. Case 1: Equilateral Triangle

The interior angle at the vertex of an equilateral triangle is $\theta = 60^\circ = \pi/3$.

Using the angular proportion:

$$ E = E_0 \times \frac{\pi/3}{2\pi} = E_0 \times \frac{1}{6} $$

3. Case 2: Regular Pentagon

The interior angle of a regular pentagon is given by $(n-2) \times 180^\circ / n$. For $n=5$, angle = $108^\circ = 3\pi/5$.

Using the same angular proportion:

$$ E = E_0 \times \frac{3\pi/5}{2\pi} = E_0 \times \frac{3}{10} $$