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Diagram

In this diagram, from what I understand the voltage between the positive terminal and the first earth ground is \$ 0V \$ and the voltage between the negative terminal and the earth ground is \$ 0V \$. So why then would you do this earth grounding? In what situation would the current flow between the earth grounds?

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The idea is that the power supply has a very high voltage output. So this diagram says "if you are afraid stupid people will come near, use a dirty trick":

You can connect the negative terminal of the supply to the mains earth ground or to the earth ground in another way and rely on it to be low in impedance in its route to the load. Then at the load side also connect the negative terminal to earth ground.

Now the current "can" flow from the power supply's positive terminal, into the load's positive terminal, through the load, into the power earth at the load's side, then from power earth back into the power supply.

Assuming we live in a world where everything always works in real life as it does on paper. This practise is largely abandoned for several very good reasons, the simplest of which is the universal rule: "You cannot rely on what you did not install yourself". Another major, but simple one is: People, that can be grounded through some point of touch, may invalidly assume they can touch the wire, because they are only touching one. The famous "one hand in the pocket" rule will not be enough any more to save people from nasty shocks, so in stead of preventing them, you are inviting them.

In my opinion a much better way of avoiding electric shock is to install the system with two good and insulated wires, as is mandatory in many parts of the world, and just not invite stupid people who like chewing on unknown wires near those wires.

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