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## Main Question or Discussion Point

Imagine that we have a charged parallel plate capacitor.

I hold a charge somewhere between the plates.

Is the electrostatic energy stored by the capacitor modified in any way?

Here is my thinking:

The total field energy between the plates is the integral of the field energy density given by:

Energy = Integral (eps_0 / 2) * |E|^2

where |E| is the total field at each point.

E = E_capacitor + E_charge

|E|^2 = E.E = (E_cap + E_charge) . (E_cap + E_charge)

|E|^2 = |E_cap|^2 + |E_charge|^2 + 2 E_charge.E_cap

Because of the spherical symmetry of E_charge I believe the integral of E_charge . E_cap is zero. Thus there is no mutual electrostatic energy between the capacitor and the charge.

The total electrostatic energy of the system is just the sum of the energy of the capacitor plus the energy of the charge.

Thus the amount of energy stored in the capacitor alone has not changed.

Is this right?

I hold a charge somewhere between the plates.

Is the electrostatic energy stored by the capacitor modified in any way?

Here is my thinking:

The total field energy between the plates is the integral of the field energy density given by:

Energy = Integral (eps_0 / 2) * |E|^2

where |E| is the total field at each point.

E = E_capacitor + E_charge

|E|^2 = E.E = (E_cap + E_charge) . (E_cap + E_charge)

|E|^2 = |E_cap|^2 + |E_charge|^2 + 2 E_charge.E_cap

Because of the spherical symmetry of E_charge I believe the integral of E_charge . E_cap is zero. Thus there is no mutual electrostatic energy between the capacitor and the charge.

The total electrostatic energy of the system is just the sum of the energy of the capacitor plus the energy of the charge.

Thus the amount of energy stored in the capacitor alone has not changed.

Is this right?