Total energy non-conservation for explicit time dependent potentials

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Total energy is not conserved when the potential energy depends explicitly on time, as changes in the potential can result in energy being transferred or transformed. For example, moving a charged particle alters the potential energy of a nearby test charge, leading to fluctuations in the test charge's energy. However, this energy change is balanced by the energy expended in moving the particle, indicating that while the total energy appears to change, it is accounted for by the work done on the system. The discussion emphasizes that energy conservation holds true overall, but the distribution of energy can vary due to time-dependent potentials. Understanding these dynamics is crucial in analyzing systems with explicit time-dependent forces.
brotherbobby
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I read from a book that the "total energy is not preserved when the potential depends explicitly on time", i.e. U(x,t). Can anyone show or prove it?

Many thanks.
 
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Total energy will be conserved. What happens is that you will be spending some energy in changing the potential U.

For instance, consider the potential energy of some test charge, T, placed near some charged particle, A. Of course the potential energy of T will change if you move particle A around. So, in that sense, the test charge's energy is changing. But anything you lose or gain will be covered by the energy used in moving particle A.
 

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