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Consider this very simple and well-known experiment with dirty/salty water:

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The light bulb turns on, and current is flowing. How can you know how this current is exactly flowing, and how the electric field looks like through the liquid? Is it a very confined path, or does it spread over the whole electrolyte? How does the conductivity of the fluid affect the shape? Are there other parameters that can tune the shape (e.g., AC frequency if not using a battery, voltage, shape of the wires and wire ends, ...)?

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Electric current generally follows electric field lines, which could be drawn by adding electric fields produced by each electrode in various points in space.

The resulting electric field should look similar to the electric field between two parallel charged wires, except of a significant fringing effect at the ends of the wires. A simplified example of such field is shown below.

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The diagram is copied from this article, which describes in detail the calculation of the current between the electrodes in electrolyte.

The shape of the field and the distribution of the current will be mostly determined by the shape and position of the electrodes.

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