In Kentucky, Mitch McConnell, the party’s Senate leader, is fending off a charismatic and wealthy conservative challenger. In South Carolina, Lindsey Graham, one of the Senate’s most reliably conservative voices on foreign policy, is being painted by primary opponents as a veritable clone of President Obama.

In Tennessee, Tea Party activists have vowed to take out Lamar Alexander, the veteran senator, former cabinet officer and two-time presidential candidate. “Senator Alexander has never been a true conservative,” said Ben Cunningham, president of the Nashville Tea Party. “His support for the amnesty bill has caused great problems for us,” he said, referring to the Senate immigration bill. “He is at best a moderate.”

Tea Party candidates have also emerged in races against Democratic incumbents in Alaska — Joe Miller, who beat Senator Lisa Murkowski in her last primary, has resurfaced — Colorado, Louisiana and North Dakota, and for open seats in Georgia, Iowa and South Dakota. Democrats hope they can benefit from a divided Republican electorate.

The Republican incumbents and party officials say they have learned from the hard lessons of the past when Tea Party candidates from the right were ignored or dismissed, only to prevail in primaries and lose in general elections. …