help with an LED light and battery setup

[vgamesx1]

Let me start off by saying I am still very new to this, but I've got the basic idea of how this works.
I need help to know what else I need and to make sure I do it right, ok I want to setup an LED strip under my desk because my keyboard slides underneath it thus cannot see it half the time, so I was thinking of getting some cheap button lights off ebay but I decided to get this instead, which runs at 12V and 0.6A (yeah, I could get a lighted keyboard but those cost $30 or more while this would only be about $10, oh and of course.. its fun to do.)
http://www.ebay.com/itm/111131398361?ssPageName=STRK:MEWAX:IT&_trksid=p3984.m1423.l2649

so my question is first how many batteries do I need? 12V worth right? so 10?
I am planning on getting some cheap ones like these http://www.ebay.com/itm/270569199363?ssPageName=STRK:MEWAX:IT&_trksid=p3984.m1423.l2649
(and yes I know they're really cheap and will actually run at about 300ma but I don't need this light to work very long a few minutes at a time and it keeps the cost low)

secondly, how many panels do I actually need to charge the batteries? I am getting some cheap 52x38mm panels and tabbing them, so do I need about 3 or 4 panels per battery? so basically for 10 batteries about 30 panels?
I also do know that I need a diode to keep the batteries from discharging into the panel, but do I need something to keep them from overcharging? or do I should I even be worrying about it at all?

lastly, about wiring, does the wire used matter much on such a small project? and I would also like to connect this to a little on/off switch, plus some tips on how to wire it all together is always welcome as haven't done this before, like should I use a PCB/breadboard to hook it up?

Thank you very much for the help, this is my first solar project and I just want to make sure I'm calculating this correctly, plus I already have ideas for a second neat little project involving the use of peltier devices.

[mariush]

The LED strip needs 12v.  It might work with less than 12, probably down to about 10v and it should work with more than 12v but you're shortening the life of the leds.

Each led should use about 20-40mA, and since you have about 15 leds in a strip, the total power should be about 300-500mA

If you want to power the strip directly from batteries, rechargeable batteries will be about 1.2v discharged, about 1.35v charged, so a pack of 8 batteries should be enough. You'd get about 10-11v which should light up the strip just fine.

But do yourself a favor and forget about eBay batteries, those are cheap for a reason. They say 3000mA but in reality they're probably 1000-1500mA if you're lucky and the chemical composition is so bad they're only going to survive probably less than 100 recharges.
If you want rechargeable batteries, I strongly suggest Sanyo eneloop : https://www.amazon.com/dp/B004UG41XW/ref=as_li_ss_til?tag=tgnethgada-20&camp=0&creative=0&linkCode=as4&creativeASIN=B004UG41XW&adid=0CNCNEH65EX1G7MNVZCX&
Yes, they're more expensive at 20$ for a pack of 8, but they're really worth it... you'll save money in time.

Alternatively, you can buy a boost regulator, which can take a low voltaje and boost it up to a higher voltage, so you could for example use 4 alkaline batteries (~5-6v) to generate 12v for the light strip.
Here's an example : http://www.ebay.com/itm/DC-3-5-30V-to-4-30V-LM2577-Boost-Module-Voltage-Step-Up-Regulated-Power-Supply-/130978297634?pt=LH_DefaultDomain_0&hash=item1e7eea3722
This takes at least 3.5v and produces up to 40v ... so if you use 4-6 non-rechargeable batteries (which have 1.5v-1.65v) then you have  6-6.6v at the input, and you can set this boost regulator to output 12v with no problems at all.
If the light strip uses 500mA, the boost regulator will use about 1A from the batteries which is a bit much, so 4-6 alkaline or rechargeable batteries will last about  one hour until they discharge.  Since you say you only need to light them up for short periods of time, that hour could last for a week or so.
By the way, with such a boost regulator, it would also be possible to just use one of those 7.5v 1A power adapters that comes with switches, scanners etc or a good usb charger that can do 1-2A (not a cheap charger, those can't give enough power)

You can improve the life of the batteries by adjusting the brightness of the leds, but now we're getting into PWM or limiting current that goes to the leds and that's another story.

[vgamesx1]

The LED strip needs 12v.  It might work with less than 12, probably down to about 10v and it should work with more than 12v but you're shortening the life of the leds.

....

Alternatively, you can buy a boost regulator, which can take a low voltaje and boost it up to a higher voltage, so you could for example use 4 alkaline batteries (~5-6v) to generate 12v for the light strip.
Here's an example : http://www.ebay.com/itm/DC-3-5-30V-to-4-30V-LM2577-Boost-Module-Voltage-Step-Up-Regulated-Power-Supply-/130978297634?pt=LH_DefaultDomain_0&hash=item1e7eea3722
This takes at least 3.5v and produces up to 40v ... so if you use 4-6 non-rechargeable batteries (which have 1.5v-1.65v) then you have  6-6.6v at the input, and you can set this boost regulator to output 12v with no problems at all.
If the light strip uses 500mA, the boost regulator will use about 1A from the batteries which is a bit much, so 4-6 alkaline or rechargeable batteries will last about  one hour until they discharge.  Since you say you only need to light them up for short periods of time, that hour could last for a week or so.
By the way, with such a boost regulator, it would also be possible to just use one of those 7.5v 1A power adapters that comes with switches, scanners etc or a good usb charger that can do 1-2A (not a cheap charger, those can't give enough power)

You can improve the life of the batteries by adjusting the brightness of the leds, but now we're getting into PWM or limiting current that goes to the leds and that's another story.

ahh, I see, Thank you this is quite helpful.

But do yourself a favor and forget about eBay batteries, those are cheap for a reason.

well yeah, I said I know that they aren't very good, but I have a new question, what about using capacitors/supercapacitors and making a capacitor bank? I would at most only need to use the lights about 30 minutes per night, (more so so 20 seconds to about 5 minutes on average) I'm not worried about their very small capacities.

[mariush]

Batteries will provide energy for a long period of time while maintaining the voltage within a small range: 1.2-1.35v for rechargeable batteries, 1.5-1.65v for alkaline batteries, 3.6-4.1v for lithium batteries.
Unlike batteries, supercapacitors store less energy, but can discharge completely ... so (for example) you can charge a 2.7v supercapacitor up to 2.69v and as you take energy out, it can get discharged down to 0v

However, if you compare the two ... let's say you have two AA batteries in series giving you 3.3v, and a 50F 2.7v supercapacitor.... if you want to power a 2v 30-50mA led, the batteries will power it for hours because the voltage will slowly go down to about 2.9v which will still be higher than 2v. But the supercapacitor will keep the led lit up for only a few minutes, and then as energy is taken out of the capacitor, the voltage goes down to less than 2v and the LED will no longer light up.
If you put let's say 5 supercapacitors in series to get 5x2.7v = 13.5v , you have the same problem : the led strip will light up just fine at 13.5v and as the energy is taken out of the supercapacitors, it will probably stop working when voltage goes down to less than about 10v. 
So again, when supercapacitor voltage will be  10v / 5 pcs = 2v on the supercapacitor, it stops working.  You're using only 20-25% of the capacity in the supercapacitor before the circuit stops working.

If the light strip needs 500mA at 12v to run, 8 AA batteries will keep it running for about 1-2 hours, maybe a bit more. You'll need probably 4000F @ 16-18v to get the same time... that's A  LOT of supercapacitors.
Now, you can get more useful energy out of supercapacitors using a better boost regulator, but still you're going to need a lot of capacitance, a lot of supercapacitors.

You really need to measure the current consumption of that light strip.. it might use less than 500mA.

I've recently made a circuit that used a 25F 2.7v supercapacitor to keep a circuit running at 5.8v and about 3-4 mA  ... it kept the circuit running for about 1 hour.  It used all the energy between 1v and 2.7v ... still only about 70% of the supercapacitor energy.
Now notice my circuit's power consumption only used 3-4 mA ... that's very little... a single led in your led strip probably uses 10-25mA, maybe more, and you have 15 leds in the strip. So you'd need a lot of supercapacitors.

It's cheaper to just use 4-8 rechargeable batteries and a boost converter to get the 12v for the light strip.... or to just plug the boost converter in 7.5v-12v power adapter and attach the led strip to the boost converter.

[vgamesx1]

hmm, just wondering but, would a boost regulator work with the output of a solar panel?

[mariush]

As long as the solar panels can provide the input current the boost regulator needs, yes.  The burst regulator takes power in bursts to produce the higher voltage at the desired current.

For example, I needed to produce 6v @ 25 mA (0.025A) from 2v ... the regulator pulled energy in short bursts of up to 0.3-0.4A so if it's cloudy or the solar panel can't produce that much current, you're screwed.