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Let's start by seeing what the largest solar cell you could practically carry hiking could get you.

Light weight solar cells are thin and fragile, but let's ignore that for now.  Let's say you can mount one on the outside of your pack that is 1x1 foot, or 0.093 square meters.  Rounding up to 0.1 m<sup>2</sup>.  Sunlight is about 1.2 kW per square meter at earth's distance.  Let's say 1 kW/m<sup>2</sup> thru the atmosphere in full sunlight reasonably well aimed at the sun.

So, at very best, you're getting 100 W onto your solar panel.  Really good solar cells, like the kind NASA can afford, are around 20% efficient.  Let's say yours is 15% to be more realistic.  Then there is some inefficiency in the electronics and in the batteries you are charging up.  Being generous, you're down to 10% overall efficiency, or 10 W usable power from your well-aimed solar panel in direct sunlight.

Just to see where we're at, lets see what 10 W does to charge a single common 18650 lithium battery.  These are usually rated for 3.7 V, and let's pick towards the high end of available capacity, so 3 Ah.  (3 Ah)(3.7 V) = 11 Wh.  Your panel could therefore charge a empty cell to full in a little over one hour.

However, that was assuming optimum aiming in full sun.  That's pretty much never going to happen on the AT.  Most of it is in the woods.  Even if you're out in an open field during a sunny day, the panel isn't going to stay aimed well at the sun.  Clouds and shade will drop the panel output several orders of magnitude.

In the end, recharging even one 18650 lithium cell from any reasonably sized solar panel while hiking the AT would take days.  You could get meaningful recharge if spending most of a sunny day in the open in a fixed location.

In the end, this simply doesn't seem worth it.  Leave the cell phone home, or recharge it the few inevitable times you have to cross civilization anyway.