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Is it possible to have electric field in water (using electrodes with voltage difference) without having electrolysis in the water (or any other reaction)?

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Well, if the voltage is lower than needed for any possible electrolysis reaction, there would be no reaction, just the field. So the answer may seem to be yes.

Now, in fact there is a catch. Once you turn on the voltage, even if the reaction is not possible, the ions will rush to the respective electrodes and form the so-called electric double layer. Its thickness, known as Debye length, is usually quite small. As for the rest of the solution, it will be effectively shielded from the electric field.

Whether or not this counts as a positive answer is up to you.

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    $\begingroup$ Voltage may be referred to potential, simply as an ability to perform work (like in French or German, for example). Only if this potential overcompensates a threshold, the electrolysis (in the meaning of discharge of ions, effective reaction of electrolysis of water) will start with noticeable, and continuous current. Below this limit, at max the mentioned "catch". $\endgroup$ – Buttonwood Oct 6 '15 at 21:33
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To complete the answer of Ivan Neretin, everywhere on earth there is a electric field around us. And if you have a glasse of water in hand, there is an electric field in it but the potentiel will be still the same in all the water then they will not have electrolysis.

For more information look at feynman lectures on physics.

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