Is it possible to charge an ebike from a 12V solar powered battery?

You can charge it, but you probably won't do as well as you think. You're on the right track but you're missing a couple of things:

• Solar panel ratings: These are optimistic and assume full sun hitting the panel at a good angle. Here's the maths. You won't get that all day, so don't expect to get as much as you would by multiplying the number of hours sunlight by the panel wattage. 12 Hours full sunlight would naively give 1800Wh, but hopefully you're out riding and not adjusting the panel every few hours, and anyway as the sun gets lower in the sky it gives you less power because the atmosphere absorbs some.

• Solar regulator: For a panel that big you really should have a regulator. Some (linear) are quite wasteful. Switching regulators are better, and can load the panel at its maximum power point for best efficiency.

• 12V battery behaviour: a lead-acid battery, even a deep discharge one, won't last for long if you drain it regularly. You've probably got about 600Wh accessible without shortening the life. However that means you don't need to produce 1200Wh/day.

• Round trip efficiency: Every step is inefficient - charging the 12V battery, discharging the 12V battery, converting 12V DC to 220V AC, converting 220V AC to 36VDC (or similar) for the battery, and charging the bike battery (also discharging the bike battery but that's not specific to this case of limited solar power).

• Other uses: I mostly plug my campervan in to the mains on a big campsite, but occasionally use a solar-charged battery for the fridge and lights. If you do the same, you'll be eating in to your battery capacity, and (slightly more efficiently) diverting some of your panel's output from charging the battery. You may of course have a dedicated panel and battery for the bike.

You'd need to choose an inverter based on the input specifications of the bike battery charger, with a little margin. Don't over-spec it too much though as efficiency will suffer - unless of course you want to run higher-powered things off it occasionally.

As for the bike battery capacity, 500Wh means 500W for 1h, 250W for 2h, 100W for 5h etc. In practice though, it will be more efficient at lower power outputs. Most e-bikes are rated to 250W because that's a legal limit in many countries (e.g. all of the EU for standard road-legal e-bikes) so you could run flat out for only 2h, but run with a little motor assistance for much longer. Hopefully you won't drain that completely, though it's not as bad for Li-ion batteries as it is for lead-acid. If you were camping at the top of a hill you'd certainly regret using too much as the available assistance reduces as the battery gets low.

There are too many unknowns to calculate properly, but I reckon you'd be close to getting a charge of the bike battery per fully sunny day, but not quite there. You'd want to monitor the state of charge of your 12V battery. If you move the camper every few days, as I do, you may be able to charge your 12V battery from the engine to ensure it's fully topped up on a day when you're driving and not riding. You should also make sure it's full before you leave home (if you're in the van for months on end that won't make much difference).

Think about how much you rely on the bike - how stuck would you be if you had a few consecutive days of bad weather?

An additional downside is electrical conversion losses.

The solar panel will likely be putting out around 12V at some current, perhaps 5W for a 500 gram panel to 20W for a 2.5 kilogram panel. That's roughly 0.5A and 2A at 12V.

Inverting that to 220V AC and then rectifying it back to DC 36V to charge the ebike battery will lose a significant percentage of the energy, leaving you with not-very-much. The best options would be:

1. Use a solar panel to keep the camper's 12V deep cycle SLA battery topped up as much as possible throughout the day. Buy and use an ebike battery charger that takes 12V input to avoid the energy losses.
2. Use your legs to power the bike, and save the battery for when you really need it. Use the battery on economy assist only, and charge it when you get back to utility power.

You might consider buying an additional battery for your bike, and swap them over if it can be removed. Then charge both/all batteries on mains.

Rather than using an inverter, get an in-car (12v) charger (they do not convert up to 220/240 AC). You might struggle with a 100Ah battery. We originally had 2 100Ah batteries with 500W panels and that worked OK, but we struggled in the winter months.