Neutral
Neutral is return path for an AC circuit which is meant to hold current in normal condition. This current could also be due to many reasons, primarily due to phase current imbalance and a few time because 3rd and 5th harmonics also.
There could also be others reasons too but the magnitude of this current is in fraction of phase current and in few case it are often even double of phase current. So Neutral wire is usually assumed to be charged (in active circuit). This neutral wire is given to ground (by grounding) to form the second terminal of neutral wire at zero potential.
The neutral wire or “grounded conductor” may be a normally current-carrying conductor, similar in some ways to a phase wire therein it'll carry an equivalent amount of current in single phase system. It is grounded (connected to the earth) at the center-tap point of transformer for variety of very complex reasons. The neutral conductor is isolated from accidental contact because it's a really dangerous normally current carrying conductor.
The reason for the neutral wire is to hold electrical currents back to the transformer during normal operations, during a manner that keeps those hazardous electrical currents faraway from people and exposed metal objects.
Earth Or Ground Wire
Earth or Ground is for safety concerns against leakage or residual currents on the system via least effort path. While phase and neutral is connected to main power wiring, earth could also be connected to body of kit or to any system which in normal condition doesn’t carry current but just in case of some insulation failure, is meant to hold some minor current.
This current isn't directly coming from live or phase wire, but is from secondary links which wasn't in-tuned with live system in normal condition. This current is typically much lesser than path current or phase current and mostly is so as of mA. But this leakage current is sweet enough to kill someone or may risk fire. Such current are being provided a coffee resistance path and sent to earth via earth wire.
Because of the difference in application we never mix grounding of neutral and earth. However both are made grounded (of-course the method could also be different). If both are going to be mixed then the world wire which isn't alleged to carry any current in normal condition , may have some charges across and can become hazardous.
The reason for the bottom wire is to eliminate any differences in potential on the exposed metal objects you would possibly touch, and to hold return currents back to the transformer in case of an electrical fault (the unintentional contact of electrical currents to exposed metal objects).
In regards to the grounding electrode connection, an easy receptacle tester wouldn't detect the presence of a ground rod, it only checks to ascertain if the bottom spade of the outlet is in an open condition (high resistance).
Last Line
In the International Electrical Code (IEC) they call a ground wire a “Protective Earth” or PE. The Neutral is called a “Neutral” or N. When the facility company only brings over normally current carrying wires (phase and neutral only), they call the neutral wire a “PEN conductor”, because it's both a Protective Earth (PE) AND a Neutral (N). They define this sort of electrical system as a “Combined” or C. Meaning that the neutral and protective earth is Combined. When you split the neutral and PE into two separate entities, they call it a Separated or S system. When the facility company bring over a PEN conductor and you split it into two in your electrical cabinet, they call that electrical system a C-S, or “Combined then Separated” system. Please see the images below for more information.
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