His humanitarian work at the Carter Center in Atlanta is widely admired; a poll conducted in May by The Atlanta Journal-Constitution found that 61 percent of the state’s registered voters view him favorably. But unlike the centrist Mr. Nunn, the former president arouses intense passions among conservatives, who detest his politics. Republicans lampoon him as a failed leader.

“There are a lot of people in Georgia, which, as you know, is a red state — has been in the past — who look with great disfavor on my administration as governor and president,” Mr. Carter said. “And I recognize that, without any hesitation.”

So does his grandson, whose broad smile and boyish looks evoke memories of Jimmy Carter as a younger man. (Senator Carter is the son of Jack Carter, the eldest of Jimmy and Rosalynn’s four children.) Like his grandfather, he is seeking the governorship after just four years in the State Senate. As he pitches himself to voters as a defender of education and the middle class, his critics say he lacks experience and is too liberal.

“He wants it both ways,” said State Representative Jan Jones, a staunch supporter of Mr. Deal. “He wants his granddaddy’s help with contributors, but when it comes to the issues, he distances himself. My guess is if his last name were Jones, you and I might not be having this conversation.”

Even here in Plains, where every detail of the former president’s life — from the peanut farm where he grew up alongside children of black tenant farmers to his brother Billy’s gas station — has been lovingly preserved, people say he might do the younger Mr. Carter as much harm as good.

“It’s great that he has his granddaddy to rely on, and certainly his granddaddy can give him lots of information,” said Jan Williams, who taught the Carters’ daughter, Amy, in fourth grade; runs the local inn, which the elder Carters helped restore; and oversees tourists at Maranatha Baptist Church, where the former president often teaches Sunday school. “But we all know that everybody is not a Jimmy Carter fan.”