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What is the electric potential in free space? Is it not zero, since there is no charge around?

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    $\begingroup$ In the free space around charges? Or in space that has no charges anywhere? $\endgroup$ – G. Smith Feb 4 at 23:36
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    $\begingroup$ In the solar wind at $1$ AU, potential difference is estimated to be roughly $400-600 \;V$ between the Sun's corona and the top of the atmosphere. See "ann-geophys.net/20/609/2002/angeo-20-609-2002.pdf". It's going to depend upon where you are in the free space - nature abhors a vacuum. $\endgroup$ – Cinaed Simson Feb 5 at 2:03
  • $\begingroup$ If there are no charges around, then it will definitely be zero. $\endgroup$ – Sam Feb 5 at 5:58
  • $\begingroup$ @Sam. There will be always some charges in free space at given moment around due to quantum vacuum fluctuations which may generate electron-positron pairs. $\endgroup$ – Agnius Vasiliauskas Feb 5 at 9:45
  • $\begingroup$ @G. Smith I meant in the absence of charges. $\endgroup$ – Karthik Feb 5 at 21:18
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Whenever you are calculating fields/potentials in a region (of interest) you are always considering the fields/potentials created inside your region. Any variation in field that you’ve calculated is a result of an effect/charge outside your region. If you want to incorporate those variations then you must extend your region to include the source of the variation.

So when we say free space, we mean that in the region we are interested, there are no charges. Thus $\vec E$ will be $0$. This means that our scalar potential $\phi$ has to be a constant since $\nabla\phi=\vec E$. But this constant can be anything as long as it gives us zero electric field. This freedom is called gauge invariance.

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  • $\begingroup$ Got it! So is it the same line of argument for why we can define the potential to be zero anywhere? $\endgroup$ – Karthik Feb 5 at 21:19
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    $\begingroup$ Electromagnetic waves, consisting of nothing but electric and magnetic fields, can travel billions of light years through space in which there are no charges (or at least not enough to significantly scatter the waves). Furthermore, they could exist in a universe unlike ours in which no charges existed anywhere. The electromagnetic field could exist completely independently of, say, the electron-positron field. $\endgroup$ – G. Smith Feb 5 at 21:38
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    $\begingroup$ Suppose I am “interested” in the “free space” one meter away from a small charged ball. The electric field is not zero there. $\endgroup$ – G. Smith Feb 5 at 21:40
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    $\begingroup$ @G.Smith My answer is limited to classical electrodynamics as that is where my knowledge currently ends. And my belief is that maxwell’s equation in free space support em waves because we live in a world with charges. Maybe I am wrong. And for the second concern, I have addressed it in my first paragraph. $\endgroup$ – Superfast Jellyfish Feb 6 at 3:36
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    $\begingroup$ @KV18, yes. Potential can be defined as 0 in free space. It’s validity lies in the absence of charges everywhere. $\endgroup$ – Superfast Jellyfish Feb 6 at 3:38

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