Why does he capacitance increase when you increase dielectric constant?

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Correct.

If you have some material between a parallel plate capacitor and apply an electric field between the plates (such as closing a circuit) the molecules of the material within the parallel plate capacitor will orient themselves according to their dipole as such:

upload_2014-11-21_1-42-49.png


This reduces the effective electric field between the plates and enhances the separation and storage of charge in the parallel plates. If the material is a good insulator it further protects from any current leaking through the weakened electric field.
 
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