Bring it along tomorrow it's hard to see from the photos.
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I believe that the capacitor on an AC motor provides a reservoir of energy to help deliver extra current on startup, when there's a spike of current demand from the motor. As I understand it inside the capacitor there is one, very large surface area conductor that is insulated from itself with a very thin insulating layer (the dielectric). Electrons flow into the conductor but can't escape due to the dielectric and so just end up hanging around in the cap until there is demand for them. The dielectric breaks down over time until it loses its insulating properties and a short developes in the cap, rendering it unable to store energy anymore!