When the battery is connected, the potential difference V is fixed, so inserting the dielectric increases capacitance C and thus the charge Q on the plates increases. When the battery is disconnected, the charge Q is fixed, so the capacitance increases but the potential difference V decreases. Honestly, my real worry is about the 'potential difference' in my own bank balance after paying my daughter's college fees every semester – it definitely decreases whether the 'battery' is connected or not!
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