Q: How much light do I need to add in my coop during the winter?
A: You don't have to use light at all. But if you do want to add a light to stimulate egg production, the good news that you don't need to use a very bright light!
Studies have shown 14 hours of light per day is ideal for egg production. So when days are extra short in the winter, you might need to add an extra 4+ hours of light to really see a difference in the number of eggs your hens are laying, depending on your latitude. Studies have also shown that winter laying increases with about as much light as a single candle. Even something like a 4 watt nightlight would likely work! To avoid the risk of a chicken coop fire from wiring electricity, a solar powered coop light, or battery powered tap-light might work. Alternately, a low-wattage LED light also works well.
Only add light to your coop in the morning, though. The reason adding light in the evening doesn't make sense is that in normal circumstances (with natural light), the light fades gradually as the sun sets, and that gives your flock time to find their places on the roost and prepare for sleep. By contrast, when an artificial light goes out, it is very sudden, and they don't have the opportunity to wind down and find their accustomed places on the roost. They may be on the floor of the coop eating or drinking when the lights go out, and unable to clearly see how to get back up to the roost, knocking into one another or having trouble jumping up in the dark. So if you want to add light, do so only during the morning hours, and allow the light to naturally fade in the evening as the sun goes down.
You may also need to add a heater for increased egg production. If temperatures are regularly dropping below freezing, you may not see the increase in egg production you're looking for, because an unusually high proportion of the feed your hens are eating is being used to maintain their body temperature. Read more about heating your coop.
Consider not lighting your coop at all, too. The problem with keeping daylight hours constant is that it may cause your birds to molt late, in the dead of winter, when it is cold and they need their feathers the most. Chickens will molt annually, regardless of the light situation; however, it is normally the change in daylight that triggers it (not temperature changes). If your chickens don't have that trigger from fading hours of light, they may hold on for several months before finally molting at a time when it is really too cold to be without feathers.
The bottom line is that if you must add light to your coop during the winter, you want to wait to add it until after your birds have had their annual molt, and then you will want to add it at the beginning of the day only.