why neutral does not shock. how can a neutral be neutral in ac current?
When I probe my city mains with one probe in the live and one in earth (which should be 0 volt) it shows a voltage of around 250 V.
That's correct - if a little high.
But when I probe the neutral and the earth it shows no voltage.
That's good too. That line has been "neutralised" by a connection to earth at your supply transformer and, depending on your local regulations, at the supply entrance to your building.
I know that current runs in one direction for 50 times in a second. So the neutral should act like live for 50 times in a second of a second[?]. Then neutral should show some voltage with the earth, which it doesn't.
No, this thinking is not correct.
simulate this circuit – Schematic created using CircuitLab
Figure 1. Voltage measurements during positive and negative mains peaks.
The mains voltage peaks at \$ \sqrt 2 \$ times the RMS voltage. For your 250 V RMS supply that will be about 350 V peak. If you had a fast-acting DC meter with max / min peak hold function you would be able to take the readings indicated in Figure 1. Neutral stays at 0 V and the live wire polarity alternates.
If you touch the neutral wire you won't get shocked but if you touch the live the wire you get shocked why and how?
Because the neutral has been neutralised. There is no voltage on it with respect to ground.
A note of caution
Since all wires have resistance there is a low voltage on the mains neutral and this increases with the current. If, for example, the neutral from your socket has a resistance of 0.2 Ω back to the fuseboard then a current of 10 A will cause the neutral voltage to rise to \$ V = IR = 10 \times 0.2 = 2 \ \text V \$.
simulate this circuit
Figure 2. The disconnected neutral wire turns live.
Also be aware that if the neutral conductor breaks and anything is plugged in to the circuit then the neutral wire may go live. Never assume that a mains conductor is at 0 V. Isolate properly.
We force it to be that way
Mains power is wired as an isolated system, with an asterisk. The asterisk came about for some very good reasons. The "safeness" of neutral is a side-effect, and an optional one.
If mains power were an isolated system (And I've run it that way, and it works), and you are grounded presumably... then it wouldn't matter if you touched pole 1 or center (I won't call it "neutral"). No current would flow. The hot and center have no relationship with earth (except through you, and with only one "wire", it's an open circuit). The system "floats".
An isolated system is exactly what you expect.
However, we build mains power to be resilient when something goes wrong. Things can go wrong with isolated systems, and one of the scariest is a transformer leak. If transformer primary leaks (even a little) into the secondary, or if there is capacitive coupling, then it de-isolates the isolated system, and "pulls it up" to thousands of volts compared to ground. Now we have a problem. In that lathe motor, coffee maker or LED light, the insulation is not rated for thousands of volts.
The equipotential bond makes the neutral
To prevent the secondary ("isolated system") from floating at high voltages, we intentionally add an equipotential bond to force a relationship to earth. You might use a transformer for the equipotential bond, e.g. in 3-phase delta (non-wild-leg) to put earth in the middle. You could also use a car battery, giving the system a 12VDC bias from earth. But usually, you use the cheapest equipotential bond available: a piece of wire. You bond one of the conductors to ground, typically "center". **Because it is bonded to earth, you label it 'Neutral'.
It really doesn't matter which supply wire you bond to neutral. Ideally you want to minimize the voltage (to earth) of the hottest hot, so the best choice is in the electrical "center" ... however, 240V wild-leg delta is an example of not doing that.
So to answer your question, neutral is cold because we made it cold.
Neutral is not quiescent; it pulses at line frequency just like the hot. The effect of the equipotential bond is to dynamically change the bias of the whole transformer secondary, to keep neutral at earth potential and make hot move away from it.
Other useful reasons
A desired side-effect of the equipotential bond is that if there is a hot-earth fault, there is a high-current path via ground wire, conduit etc. back to the neutral-earth equipotential bond, and ultimately back to neutral. This completes the circuit, allows high current to flow, and causes a circuit breaker trip, which arrests the ground fault. Remember, current wants to return to source, not to ground. It doesn't care about ground, except that the equipotential bond makes it care.
For a variety of reasons, there needs to be exactly one equipotential bond. Another one would create redundant (paralleled) paths for normal neutral (return) current, and that causes all sorts of mischief.
The neutral does not act like the live because the neutral is tied to ground at a single point somwhere.
You are imagining the neutral and live both move above and below each other about ground. Relative to each other, the neutral and live do move above and below each other. This is if you think of them in complete isolation relative to each other and only each other.
In the real world there are more absolute voltage potentials like ground or earth. If left to float, the live and neutral move together above and below some common-mode voltage. But in the real world we tie the neutral to ground for safety purposes. We don't want the live and neutral floating to whatever common mode voltage they feel like. The equipment probably wouldn't care and would still function since it only cares about what live and neutral are relative to each other, but it's dangerous to you (who is at ground potential) if their common mode voltage floats up to 1kV relative to ground.
Since the neutral is tied to ground somewhere, then the neutral becomes more or less fixed to ground. In that case, live line ends up doing all the movement above and below the neutral.
That said, you can still get shocked by the neutral if there is a load. The neutral is tied to ground SOMEWHERE but that somewhere might be quite far from where you are looking at the neutral. That means there is an impedance between where the neutral connects to ground and where you are looking at the neutral.
So if there is no load current then nothing disturbs that impedance and the neutral sits at ground. But if there is a load current, then the voltage developed across that impedance can be enough to have the neutral voltage rise above ground to shockable levels when there is a load current.