RICHMOND, JAN. 21 -- Lawmakers from tobacco-growing areas were left smoldering today after a bill requiring no-smoking areas in all public places in Virginia, including restaurants, schools, hospitals, libraries and museums, was introduced in the General Assembly.

The bill, sponsored by Del. Bernard S. Cohen (D-Alexandria), a former cigarette smoker, and Sen. Thomas J. Michie Jr. (D-Charlottesville), who smokes a pipe, was given little chance of surviving the challenge of Virginia's powerful tobacco lobby.

"A major portion of our economy comes from the manufacturing of tobacco products . . . . I'm going to be opposing the bill," said Del. Mark W. Hagood, a freshman Republican from Southside Virginia, near the North Carolina border.

Said Sen. Clive L. DuVal 2d (D-Fairfax), a cosponsor of the bill: "Tobacco is king in Virginia. {The bill} hasn't got too much of a chance."

Formerly the core of Virginia's economy, tobacco remains the state's biggest cash crop, accounting for $207 million in annual sales, about a third of the sum for all farm products. Virginia is the nation's fourth leading tobacco-growing state, behind North Carolina, Kentucky and Tennessee.

Nonetheless, Cohen and Michie said that mounting evidence of the harmful effects of "secondhand" tobacco smoke on nonsmokers gives the legislation a fighting chance.

"When is Virginia going to meet its obligation to protect the health of its citizens?" Michie asked in a news conference.

The bill does not specify where and how large the designated no-smoking areas in public places should be, other than to say that the restricted areas should be "reasonably substantial."

The bill would ban smoking altogether in elevators, school buses, restrooms, hospital emergency rooms, doctors' and dentists' waiting rooms, jury deliberation rooms, polling rooms and city or county health offices. Private employers also would be required to designate no-smoking areas.

A number of localities around the state have ordinances that require designated no-smoking areas, particularly in large restaurants. Fairfax County enacted such a law in December 1985, and Arlington followed.

In Maryland, Gov. William Donald Schaefer signed an executive order last year requiring state agencies, airports and public universities to ban smoking in public areas.

Nationally, concerns about the impact of smoking on health and productivity have prompted a growing number of private employers to impose restrictions on when and where employees may smoke.

Michie and Cohen said 39 states have a similar law in place, and stressed survey results showing that a large majority of people favor the designation of no-smoking areas in public places.

However, when Michie rose on the floor of the Senate today to announce he was introducing the bill, Sen. Richard J. Holland (D-Windsor), who represents a tobacco-growing district in Southside, immediately lit up a cigarette. He later said of the bill, "I don't think it'll pass."