PX157 - B8e - potential difference between two oppositely charged plates

planes have charge densities, $+ σ$ and $- σ$ , and are separated by a distance, $d$
origin lies on the bottom plate
for the top (positive) plate:

{\vec{E}}_{t o p} = - \frac{σ}{2 ϵ_{0}} \hat{z}

{\vec{E}}_{t o p} = + \frac{- σ}{2 ϵ_{0}} \hat{z}

\vec{E} = {\vec{E}}_{t o p} + {\vec{E}}_{b o t t o m} = - \frac{σ}{ϵ_{0}} \hat{z}

V_{a} = \int_{d}^{0} - \frac{σ}{ϵ_{0}} \hat{z} \cdot \hat{z} d z = \frac{σ d}{ϵ_{0}} = E d