The energy of a charged capacitor is given by , where is the charge of the capacitor and is the potential difference across the capacitor. The energy of a charged capacitor can be described as the energy associated with the electric field created inside the capacitor.
In this problem, you will derive two more formulas for the energy of a charged capacitor; you will then use a parallel-plate capacitor as a vehicle for obtaining the formula for the energy density associated with an electric field. It will be useful to recall the definition of capacitance, , and the formula for the capacitance of a parallel-plate capacitor,
, where is the area of each of the plates and is the plate separation. As usual, is the permittivity of free space.
Find the energy density of the electric field in a parallel-plate capacitor. The magnitude of the electric field inside the capacitor is .
Express your answer in terms of and appropriate constants.
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