How many hours of north shadows on day of summer solstice?

Question

If you stand in the middle of the continental USA and draw an imaginary east west line, on the day of the summer solstice, the sun will appear to be north of that line in the early AM and late afternoon. How many hours will the sun appear to be south of that line, such that the shadows will point at least partly north?

As the calendar leaves the solstice, how fast does that answer change?

Answer 1

The answer depends strongly on your latitude. Here are approximate numbers, ignoring the effects of refraction and the size of the solar disk:

Latitude 45 N: 3 hours 25 minutes, morning and evening, sun is north of the east-west azimuth; total time north of the line during the day is twice that, or 6 hours 50 minutes. Between these times, the Sun will spend 8 hours 35 minutes south of the line. I will write this as:
Latitude 45 N: 3:25 / 8:35 / 3:25
Latitude 40 N: 3:30 / 7:51 / 3:30
Latitude 35 N: 3:43 / 6:54 / 3:43
Latitude 30 N: 4:13 / 5:29 / 4:13
Latitude 25 N: 5:19 / 2:54 / 5:19

South of the Tropic of Cancer, the Sun is north all day long at the June solstice.

Answer 2

What an interesting question. At first glance it seems like it would be really hard to answer, requiring lots of spherical trigonometry calculations. I can't answer it off the top of my head but may come back later and add on to this answer if I come up with anything.

Answer 3

It depends on the exact latitude, and involves some fairly messy spherical trigonometry. You will need to delve into a trigonometry text to work this out.