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Short explanation of this phenomenon: Electric field lines like to stay in water, which has high dielectric constant. Breaking the water bridge would force them to go through air, which is much more energetically costly.
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When a high voltage is applied to pure water filling two beakers kept close to each other, a water bridge forms bit.ly/2wafXZb
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This is not about least action; there is no real dynamics here. It's just straightforward energy minimization. "Electric field lines stay in water" just means that water molecules align in the presence of electric field. This alignment has the effect of focusing field lines.
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