Why don't farmers add carbon to grow crops

But they have to add nitrogen and phosphorous through fertilizer?

Because plants take the carbon dioxide from the air and make it into carbohydrates (sugars). Nitrogen and phosphorus are needed in much smaller quantities than is carbon and the point of entry is usually through the roots on land-based crops, though legumes (beans) can fix nitrogen from the air.

Carbon for nutritional purposes is provided by the air as CO2.

Organic carbon is very important for crop growth - this is the bulk of what soil is (especially clays). So carbon is added - as manure and compost, though carbon is never the primary purpose. Just letting the plants grow also builds up carbon, while tillage removes it. It's also not really economically to add more top soil to a field (greenhouses are a different matter).

Organic carbon is important for soil and nutrient retention. Sandy soils (low in organic carbon) dry out much faster and have a lot of runoff problems.

They often do add when they grow in greenhouses. Carbon is absorbed in plants as the gas CO2. They often increase the concentration to about 1000 ppm which greatly increases the yields of many crops. Nitrogen and phosphorus are dissolved in various compounds when plants absorb them generally through the roots.

Quote:

"Plants get their carbon from the air (CO2) so adding carbon to the soil wouldn't help plants much."