# Charge density (σ) and distance between plates of capacitor given just the relationship between potential difference (V) and width of a metal slab?

I’m trying to become an electrical engineer and I was given a problem in school (calculus-based physics) to work on. Basically, a parallel plate capacitor is charged to the same potential difference as the battery that is connected to it. Then, a metal slab was inserted between the two plates (after the battery is removed) and the new potential difference (V) across the capacitor was measured. Next, various metal slabs with different thicknesses (cm) replace the original, again measuring the new potential difference across the capacitor. I was instructed to graph potential difference (V) as a function of slab thickness and got a linear relationship with a slope of about -2.5 V/cm.

Now, I’m tasked with finding charge density (σ), which is charge/area, and the distance between the two plates. My professor insisted I must use the slope to do so. How do I do this?

To be quite honest, I’m quite lost. I’m not too familiar with the capacitance equations but I know that Charge = Capacitance * V and Capacitance = KEoArea/distance, even though I’m unsure if these relate. I derived V without the metal slab to be σd/2Eo but I don’t think I can use this with a metal slab now inserted. Any help is appreciated! Hopefully someone will be able to find the answer!