How long do the sun rays stay at the Tropic of Capricorn at the Solstice?

I assume you're thinking of the Sun being directly overhead there at the solstice, so its rays come down vertically there. Of course the Sun shines there every day, just at different angles. So I'm having to guess at what you meant.

If that's what you meant, that's only for an infinitely tiny instant, as there is constant motion. As the Earth continues along its orbit, the place where the Sun is directly overhead so some of its light comes down vertically starts to move back north immediately.

If you plot how high the Sun is in the sky at noon against time, you will get a graph that looks like a sine wave. Up and down once per year. Plot what latitude the Sun is overhead at noon against time, and you get the same shape, going up and down between the Tropic of Cancer and the Tropic of Capricorn. The Tropics are tangents to the curve and only JUUUUSST touch it for a very tiny moment.

You've hit on an interesting point in mathematics. The mathematics to be able to actually deal with tangents and rates of change wasn't even invented until the 17th century. Sir Isaac Newton invented it and called it his "method of fluxions", but so did Gottfried Leibniz, and with Sir Isaac's tendency to not tell anyone what he was doing, nobody knows who was first! Now we call it calculus and use Leibniz's notation for it.

This is so advanced that you probably don't even do it in school, or if you do, not until you're 16 or 17 like I did. It was only when I did A level maths (yes, I'm English, the American equivalent would be AP math) that we really started learning calculus and all that stuff about differentiation and integration. It was when I did second-order differential equations in three dimensions during second-year university physics that my poor brain started to implode...

Anyway, what calculus tells you is the point where the curve of the Sun's apparent movement touches the imaginary line is infinitely small. They touch, but for what is, mathematically, zero time.

It changes instantly. The Earth does`nt stop spinnining at all.

No they turn right back

This year on the 22nd it was at 12 Midnight

And today the 23rd December we had 6 Seconds more Sunshine

I don't know about you, but I went to the Beach

As soon as the sun reach at Lat. 23.5 degrees South ,then its start climbing up to 23.4 so on and so forth up to equator,continuously non stop.

But one phenomena which is interesting to point out is ,when the sun reach Lat. 23.5 degrees South and its location is at the Mid day of Hawaiian Islands, in the Mid of Pacific Ocean , the whole world of all the Continents would become dark or in the night time and before dawn for a certain period of time.

The DIRECT (right-angle) rays of the sun are at the Tropic of Capricorn only for an instant. That's the definition of that line. But you would need some high-precision math to determine the difference between that instant, and 12 hours. The sub-solar point moves only 554 meters north-south in that time, on that date.

How long is a sine wave at it's peak? Only an instant.

All Day 365 days a year.

Just at different angles.

Simplistically, the solstice lasts about one second long (at any given spot). However, it is worth noting that the suns rays strike the Tropic every day of the year -- just at an angle. So, I'm not sure what you are asking here.