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If we have two charged particles in an electric field, then there is a potential energy in the system.

I am aware that it is meaningless to talk about the potential energy of a single particle alone. However,what about the phrase 'potential energy of charge $q_1$ due to $q_2$', for example, does that make sense?

I just need a bit of elucidation on the phrasing we use when we talk about potential energy (and some explanation of why would be appreciated).

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  • $\begingroup$ Just to clarify, so you have two charges plus an external electric field? And you want to give different names to different parts of the potential energy: potential energy of $q_1$ in the external field, potential energy of $q_2$ in the external field plus a term which depends on both $q_1,q_2$ $\endgroup$
    – user1583209
    Commented May 10, 2017 at 12:35
  • $\begingroup$ It is the electric potential energy of the system of two charges. $\endgroup$
    – Farcher
    Commented May 10, 2017 at 12:40
  • $\begingroup$ @user1583209 The electric field is created by the charges themselves. Is it wrong to speak about the potential energy of q_1 due to q_2 $\endgroup$
    – PhysicsMathsLove
    Commented May 10, 2017 at 12:59
  • $\begingroup$ I see (might want to edit your first sentence in the question which reads as if there is an external electric field). Personally, I'd describe this situation by saying that one charge (e.g. $q_1$) creates an electric field and the other charge ($q_2$) has a potential energy in this field. I would be fine with your short version (potential energy of $q_1$ due to $q_2$ as well. Also, saying just "potential energy of the system of two charges" (suggested by Farcher) is fine and avoids making the distinction between field-creating charge and "test"-charge. $\endgroup$
    – user1583209
    Commented May 10, 2017 at 14:03

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Imagine it this way (strictly intuitive and physical):

  • If both charges 1 and 2 are positive, there will be some repulsive force between the two, i.e. charge 2 gets repelled owing to the presence of the charge 1 and vice versa.

  • In case the latter was absent, there would be no such force.

  • With both these charges present, there would also be some energy of interaction. Imagine this as a manifestation of the force, if the force "pushes" the other charge away, its energy also increases. Notably, this energy wouldn't have "existed" if the other charge was missing, so this is really "due to the presence of the other charge".

  • When you set up any charge configuration, e.g. a system of 100 electrons (i.e. 100 units negative charge) on a conducting sphere, you are doing work against this repulsive force, and therefore expending certain amount of energy in setting up this charge configuration. e.g. when you add the first electron, there's no problem, but the second electron experiences a repulsion because the first one is already present there earlier. Likewise, the third experiences even more, there are two electrons repelling it, and so on.

This is the physical meaning of the potential energy of $q_1$ due to $q_2$, or rather, the potential (energy) of any system of charges.

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