Prime Minister James Callaghan announced today that the national election forced by the fall of his government last night will be held on May 3.

The defeated Laborite prime minister made the announcement after consulting this morning with the queen, who formally set April 7 for dissolution of the outgoing House of Commons. By a vote of 311-310, it had voted no confidence in Callaghan.

The next Parliament is to convene May 9, six days after the election.

That leaves five weeks sandwiched around Easter for voters to decide whether they want Callaghan or Conservative Party leader Margaret Thatcher to tackle Britain's problems of high prices, unemployment, low productivity and declining world influence.

Despite disclaimers from both sides, it is beginning as a bitterly ideological and personalized campaign, pitting the current unpopularity of the Callaghan government's economic performance against the relatively low personal popularity of Thatcher.

In a series of speeches culminating in her successful demand last night that the House of Commons support her motion of no confidence, Thatcher has seized on the frustration with a winter of frequent strikes and rising inflation that pulled the Labor Party down well below the Conservatives in public opinion polls.

Promising "less taxes and more law and order" if she becomes prime minister, Thatcher declared last night that "law and order refers not only to vandalism and violence. It also means our citizens expect, and are not getting, an ordered or orderly society. They expect rubbish to be collected, sschools to be open and hospitals to be functioning, and they are not."

She is attacking, as she has throughout her four years as opposition leader, what she sees as runaway big government, rampant nationalization of British industry, high taxes that rob industry of profits and workers of incentives, unchecked labor union power and entrenchment of socialism during Labor's control of Parliament for 12 of the last 15 years.

The public opinion polls charting Labor's loss of support since its peak of popularity last October show that many voters agree with Thatcher's complaints about Labor.

But the polls also show that Callaghan is still much more popular personally with the voters than his government, while Thatcher is significantly less popular than her party or her shadow cabinet. In particular, the polls show that a preponderance of voters of every age and background believe she is "the mose extreme" of candidates considered.

So the underdog Labor Party is offering Callaghan as an experienced leader with global experience who will steer a steady course, while portraying Thatcher as inexperienced, strident and given to simplistic solutions for complicated problems.

Callaghan has referred frequently in recent weeks to his representation of Britain at numerous summit conferences and to the two years of success he had-before this past winter-in reducing inflation by persuading union leaders to agree to a policy of wage restraint.

"It would do great harm." Callaghan warned in a nationally televised speech today, if the country were "jerked in the opposite direction." He said the problems of prices and unemployment could be solved only by "co-operation and partnership, and not confrontation" or by "having an upheaval in industry and with the unions."

Callaghan 67, is a life-long trade unionist who never went to college. He started work in a government tax collection office at 16 and has been in Parliament since 1945. He served in three top Cabinet posts in Labor governments-chancellor of the exchequer, home secretary and foreign minister-before becoming prime minister after Harold Wilson's sudden mid-term retirement three years ago.

An effective, witty and frequently caustic speaker, Callaghan has the reputation of being a crafty parliamentarian who bargained successfully for the support of various minor parties to keep his embattled government in power until last night's vote.

Thatcher, 53, is a butcher's daughter who studied chemistry and was elected president of the Conservative Union at Oxford before becoming a tax lawyer. She is a studious politician who compensates for an uninspired speaking style with hard work and tough infighting that defeated party rivals rivals on both the left and the right of her traditional conservative position.

Labor leaders, however, depict her as a staunch right-winger whose abrasive, doctrinaire manner has alienated an important wing of her party, including the last conservative prime minister, Edward Heath.

There also have been recurring signs, despite strong public admonitions from Callaghan to the contrary, that the fact that Thatcher is the first woman with a chance to become prime minister of Britain-or of any European nation-may become a factor in the campaign.

After a Labor Cabinet minister made a strong personal attack on Thatcher over the weekend, it was leaked to the newspapers that Callaghan later told a meeting of junior labor ministers he did not approve of campaigning against Thatcher's personality or sex.

Then, in las night's Commons Debate, a Labor back-bencher said that "ever since she became leader of the Conservatives, [Thatcher] has had everything done to her and for her except a facelift" and that her "harsh metallic voice has now been replaced by an artificial sexual huskiness."

This led Callaghan to declare in Commons today, after he announced the election date, that "I trust this election will not be fought on anyone's personality-but on the issues with which we are confronted."