I am perusing through my principles of electromechanics textbook and was puzzled by how the electric field energy equation was formulated. For example, the Electric field energy for a capacitor is defined by the integral of voltage (lambda dot in the attachment) as a function of charge (the prime symbol indicates that the charge is also a function of time but the charge coordinate is used instead). If you reverse the placement of voltage and charge in the equation, you obtain coenergy, which isn't a physically significant value, but is useful for evaluating energy when the units are more convenient. However, that does not explain why the order matters, but I did think about it. I listed my thought process below:
The main differential equation for a capacitor is $I = C{dV\over dt}$, where I=current and V=voltage. If you take the integral of each side with respect to time, you obtain, ${1\over C}\int dt=V$. If you take the integral of current with respect to time, you obtain charge q if current is constant over time and since voltage is the dependent variable and charge is the independent variable, energy is a product of the changing charge, not the changing voltage, since it is the charges themselves that create the electric field. That is my reasoning, but I don't know if I am close or really far off.
I posted all relevant attachments that I could think of below. I would be very grateful for your help. Thank you plenty!
Best, Truth_Seeker24
