- #1
pmr
- 30
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I'm currently taking an EM course. We're doing AC circuits, but I'm having a hard time understanding how voltage is defined.
In the electrostatic context I understand how the voltage is defined as the line integral of E over a path, and I get how that line integral has nice path-independent properties because E has no curl. But then our course introduces magnetism, and circuits. The sort of circuits we're looking at have inductors, or lots of self-inductance, so even magneto-statics goes out the window.
When I ask about how voltage is defined in this more complicated context I get lots of answers like "voltage is still just the integral of E * dl", but I'm not happy with that because it's path-dependent, and therefore meaningless unless a convention is chosen for taking the path (which I've never seen done), or unless an argument is given for why that detail isn't important (which I've also never seen done).
I also sometimes get answers like "voltage is just the difference in potential energy divided by charge," but I'm also not happy with that because people giving this answer uniformly fail to answer the follow-up question of "what is potential energy?" You can't define electric potential energy in usual way, for the reasons I just explained in the last paragraph. And anyway there are some choices of gauge for which the [itex]\phi[/itex] in [itex]\vec E = -\nabla\phi - d{\vec A}/dt[/itex] is declared by fiat to be [itex]\phi=0[/itex] everywhere and at all times (which has interesting consequences for how [itex]\vec A[/itex] has to be defined, but nonetheless renders the concept of the scalar potential null and void).
Lastly, I get a lot of people saying that I'm overcomplicating things. But all I'm asking is for some definition of this pervasively used concept of "voltage" that doesn't have big gaping holes in it. I don't think that's too much to ask for, and I think it's worth nitpicking about. If there are assumptions or approximations going on that render my nitpicking extraneous, then I want to know exactly what those assumptions or approximations are.
Also, to the admins: I wasn't sure where to post this question. If I'm in the wrong section then feel free to move me.
In the electrostatic context I understand how the voltage is defined as the line integral of E over a path, and I get how that line integral has nice path-independent properties because E has no curl. But then our course introduces magnetism, and circuits. The sort of circuits we're looking at have inductors, or lots of self-inductance, so even magneto-statics goes out the window.
When I ask about how voltage is defined in this more complicated context I get lots of answers like "voltage is still just the integral of E * dl", but I'm not happy with that because it's path-dependent, and therefore meaningless unless a convention is chosen for taking the path (which I've never seen done), or unless an argument is given for why that detail isn't important (which I've also never seen done).
I also sometimes get answers like "voltage is just the difference in potential energy divided by charge," but I'm also not happy with that because people giving this answer uniformly fail to answer the follow-up question of "what is potential energy?" You can't define electric potential energy in usual way, for the reasons I just explained in the last paragraph. And anyway there are some choices of gauge for which the [itex]\phi[/itex] in [itex]\vec E = -\nabla\phi - d{\vec A}/dt[/itex] is declared by fiat to be [itex]\phi=0[/itex] everywhere and at all times (which has interesting consequences for how [itex]\vec A[/itex] has to be defined, but nonetheless renders the concept of the scalar potential null and void).
Lastly, I get a lot of people saying that I'm overcomplicating things. But all I'm asking is for some definition of this pervasively used concept of "voltage" that doesn't have big gaping holes in it. I don't think that's too much to ask for, and I think it's worth nitpicking about. If there are assumptions or approximations going on that render my nitpicking extraneous, then I want to know exactly what those assumptions or approximations are.
Also, to the admins: I wasn't sure where to post this question. If I'm in the wrong section then feel free to move me.
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