On Capitol Hill it's known as the Brazilian blunder, and in just four weeks it has not only embarrassed Rep. William V. (Bill) Alexander Jr. (D-Ark.) at home but undermined his hopes of rising further in the House Democratic leadership.

Alexander's August sojourn in Rio de Janeiro is the latest episode in the crowded race for House Democratic whip, the No. 3 job in the Democratic line-up and the only leadership post that still appears up for grabs. Leadership posts will not be chosen by the House Democratic caucus for 15 months, but the jockeying for whip is under way, pitting California Democrats against New Yorkers, the Midwest against the South, liberals against moderates and conservatives.

And in as intense a competition as this has become, said one lawmaker, "people are always looking for an excuse to eliminate someone. Bill Alexander just gave them the excuse."

Alexander is the House member who requisitioned an Air Force plane from the Pentagon during the August congressional recess to fly what he said would be a delegation of five lawmakers to Brazil. But when the aircraft took off, Alexander, the chief deputy whip and fifth-ranking House Democrat, was the only member of Congress aboard. The others said they never told Alexander that they definitely would make the trip.

The weeklong trip, which cost the federal government about $50,000, has produced a spate of scathing editorials and cartoons in Arkansas newspapers and rekindled GOP interest in trying to knock the nine-term Alexander out of his secure rural eastern Arkansas seat. The Republicans filmed the aircraft returning to Washington.

"The impression now is he is vulnerable," said longtime political observer Ernie Dumas, an editorial writer with the Arkansas Gazette. "The common assumption is it was just an unwarranted trip . . . . It seemed to leave a bad taste in people's mouths."

In Washington, the Brazilian trip provoked stern public criticism from House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.), angered lawmakers who felt it made them all more vulnerable to charges of "junketeering" and caused wide speculation that Alexander may have to drop out of the whip's race to avoid a humiliating loss.

Alexander, a telegenic 51-year-old moderate, has predicted that his trip to study alcohol-based fuel production in connection with a study at the University of Arkansas will help him in the whip's race by showing his colleagues that he can "take the heat."

"For members of Congress who, like me, know the importance of the trip . . . especially those who know the importance to my state, they applaud the trip," he said this week. "Those who do not, question it because of the presumption created by the media."

Alexander supporters are less sanguine. Said Rep. Frederick C. Boucher (D-Va.), who is working with Alexander, "Obviously, it hasn't helped him" but predicted that Alexander would recoup before the vote.

Another Democrat, who spoke on the condition that he not be named, said, "Bill got damaged badly. A leadership guy is supposed to take the heat, not generate it."

Lawmakers in all Democratic camps said that even before the Brazil trip, Alexander seemed to be having some trouble.

There were snickers last spring about the way Alexander was featured in the opening anecdote of a Wall Street Journal article about lobbyists. The story had him brandishing an African spear and saying "boogaloo" during the visit of a lobbyist to his office. (Alexander maintains he did not say "boogaloo" when asked to show the spear but used the word "jambo," which he said is a greeting in an African dialect. Nonetheless, in at least one Arkansas newspaper, he now is called "Boogaloo Bill.")

In a race that is decided not by issues or position papers but by personality, careful political calculation and one's ability to mix into the "good-old-boy" atmosphere of the House Democratic cloakroom, Alexander is known as a loner.

He is widely respected for his strong leadership on certain issues, such as going after President Reagan's Central American policies and countering the barrage of Republican after-hours televised floor speeches by organizing the Democrats to offer up their own.

But unlike nearly all his opponents in the whip race, Alexander does not constantly work the floor, corralling Democrats to ask for support, listening to colleagues' complaints and massaging their egos.

In addition, coming from a state with only three Democratic House members, Alexander does not have the large natural political base of either Rep. Tony Coelho (D-Calif.) or Rep. Charles B. Rangel (D-N.Y.), widely regarded as the front-runners in the race. Alexander has some southern backing but, unlike most members from the conservative South, he often votes with the more liberal Democratic leadership.

He will not disclose how many firm supporters he has, saying only that he is "very encouraged" by his backing, even since the Brazil trip. Boucher said he thinks that it is 60 members, but other Democrats familiar with Alexander's effort say he probably has no more than about two dozen commitments.

Coelho, 43 and a leader of the large group of younger Democratic lawmakers, is said to have fairly solid support among California's 27 Democrats. As head of the Democratic Congressional Campaign Committee (DCCC), Coelho also can count on close relations with many of the new Democrats who may be elected to Congress in 1986 just before the whip vote. The DCCC post also puts Coelho in a position to do favors for many colleagues.

Widely respected for having rejuvenated the DCCC and developed the political strategies that helped protect the Democrats in the 1982 and 1984 elections, the tireless Coelho nonetheless has a sizable group of detractors, known as the ABC caucus -- "Anybody But Coelho." They say the four-term California congressman is too political and ambitious.

Coelho's forces have put out the word that he already has more than 110 commitments.

Rangel, 55, is backed by most of the 19-member New York Democratic delegation and the 20-member Congressional Black Caucus. In addition, he is a subcommittee chairman on the House Ways and Means Committee and may be able to win support from the committee's powerful chairman, Rep. Dan Rostenkowski (D-Ill.). Rangel's work in Ways and Means on cigarette tax legislation has also earned him backing in southern tobacco states, according to his supporters.

Rangel is also the only candidate from the Northeast. Lawmakers there say that if House Majority Leader James C. Wright Jr. (D-Tex.) becomes speaker and if Rep. Thomas S. Foley (D-Wash.) becomes majority leader, as expected, Rangel would be needed as whip in 1986 for "regional balance."

However, lawmakers said Rangel may be hurt by the fact that another liberal northeastern black, Rep. William H. Gray III (D-Pa.), recently was elected Budget Committee chairman by the caucus and the fact that some Democrats would oppose giving another black a high leadership position.

A Rangel supporter said the Harlem congressman has about 65 firm commitments.

Rep. Martin O. Sabo (D-Minn.), 47, and Rep. W. G. (Bill) Hefner (D-N.C.), 55, the only other officially declared candidates, lag far behind Coelho and Rangel, according to Democrats.

Sabo, a liberal former speaker of the Minnesota House of Representatives, has strong backing from midwestern lawmakers and from the large liberal wing of House Democrats. Hefner, who suffered a heart attack this year, is known as one of the most affable members of the House and has a strong southern base.

The two agree they are the underdogs but remain hopeful."I remember Jim Wright's race for majority leader that he won by one vote," said Hefner, referring to 1976 surprise victory of Wright over the late representative Philip Burton (D-Calif.). "That thing went on longer than a dead snake."