WASHINGTON  After subjecting the nation’s biggest banks to the most public scrutiny in decades, federal regulators ordered 10 of them on Thursday to raise a total of $75 billion in extra capital and gave the rest a clean bill of health.

The long-awaited results of the “stress tests” set off an immediate scramble by major institutions for more capital. By June 8, they must give regulators their plans for raising the money, and raise it by November.

The verdict was far more upbeat than many in the industry had feared when the tests were first announced in February. And the banks that came up short will have to raise much less than some analysts had expected as recently as a few days ago.

The stress tests were aimed at estimating how much each bank would lose if the economic downturn proved even deeper than currently expected. But regulators gave the banks a break by letting them bolster their capital with unusually strong first-quarter profits and also by letting them predict modest profits even if the economy again turns down.