U.S. economic aid to Egypt has grown so extensively and swiftly, becoming the largest such program in the world, that half the money earmarked for the Egyptian economy since 1975 has yet to be spent.

The backlog reflects the difficulty of funneling almost $1 billion a year through the sluggish Egyptian bureaucracy and government-owned businesses in an effort to ease the effects of years of economic neglect and mismanagement.

It also underscores the obstacles faced by Egyptian officials and the big U.S. aid mission here to show that peace brings prosperity. That economic imperative was laid down by President Anwar Sadat and the Carter administration for basically political reasons revolving around the peace treaty with Israel.

The stack of unspent funds has reached about $1.9 billion, excluding another $1.5 billion in military aid and $300 million in extra exonomic aid promised this spring as part of President Carter's bargain with Israel and Egypt that helped persuade them to sign the treaty.

It includes about $43 million agreed to as far back as 1975 and still awaiting expenditure. The $372-million aid program that year marked, for instance, the start of a project to build two grain silos in Alexandria and Cairo that Egyptian officials say will be finished in 1980, five years after the decision to build them.

"You should see how time-consuming it is to find out the best site for one of these in Alexandria," the minister of state for economic cooperation, Gamal Nazer, said in an interview.

Donald Brown, who heads the aid mission here, said the total backup of unspent money is not extraordinary viewed in light of the huge amounts involved. Overall, the United States has pledged more than $3.7 billion in economic assistance to Egypt between 1975 and the end of last month.

In addition, what began as a "quickshot" aid program, launched as part of former secretary of state Henry Kissinger's shuttle diplomacy after the 1973 Middle East war to bring Sadat into U.S. peacemaking, has evolved into a broad, long-term effort to transform Egyptian economy.

As a result, the United States is involved in aid projects ranging from birth control to sewage lines and from telephones to management of the bureaucracy. Many of these projects are long-term efforts that take a number of years to complete or even to get under way.

"It's in the projects that there is a big gap between commitments and expenditures," Brown said in an interview.

Simple commodity imports can go much faster. Of the approximately $539 million committed through June 30, 1979, for example, more than $418 million has been spent, a percentage sharply above the average since 1975.

This is because most of it has gone to finance wheat and flour imports designed to keep down the price of bread despite inflation estimated unofficially to be running at about 25 percent a year.

Projects, on the contrary, require the Egyptian government to follow a long, complicated procedure before the money is spent. This includes negotiating with a consultant, getting his report, asking for and choosing a design, selecting engineers and getting their blueprints, taking construction bids and then finally beginning the actual work.

In Egypt, which often seems to be suffocating under the weight of its ponderous disorganized bureaucracy, the process takes several years. Even for a direct import it took 18 months just to negotiate specifications for the first batch of buses purchases with U.S. aid and constructed in the United States, Nazer said. It has taken up to a year to negotiate contracts with private U.S. consultants asked to say whether a prlject is worth undertaking in the first place.

The process sometimes is lengthened by differing points of view between U.S. aid officials, who insist on U.S. feasibility studies and tight controls, and their Egyptian colleagues, who often are shocked at priced charged by U.S. firms or who feel Egyptian engineers could do the job just as well.

"The implementation time will be substantially faster from now on as we get used to each other," Brown said.

Many projects on which negotiations have been moving laboriously ahead during the past several years are approaching the final stages. As a result, Brown predicted, the gap between money committed by Congress and money actually spent in the Egyptian economy will narrow, until by 1980 or 1981, "we will be spending about what we are committing."

To run the massive program, the aid staff here has grown to 120 positions, with about 90 presently filled. The total U.S. Embassy staff is about 175 persons.

One of recently departed ambassador Hermann Eilts' last public acts here was to warn in an interview about allowing the giant aid program to foster a similarly giant embassy staff that could make U.S. involvement in Egypt too obvious for comfort.

His successor, Ambassador Alfred Atherton, is known to want to limit the embassy staff as much as possible. Sadat is reported to be determined not to repeat Egypt's mistake with the formerly pervasive Soviet involvement here.

But U.S. officials already are involved in projects as intimate as birth control, with all its social overtones in Egypt's Moslem society, and as fundamental as farming, which has changed little along the Nile since ancient times.

In addition, a team of about 80 U.S. specialists will be attached to the Egyptian Air Force to train Egyptians in caring for 35 F4 Phantom jets scheduled for delivery beginning this fall as part of a $1.5 billion military allocation.

Other military specialists are likely to follow if, as expected, the military aid later includes ships, sophisticated missiles and armored vehicles.

The F4 technicians are coming without families. U.S. officials emphasized that their stay is designed to be temporary. But their very presence within a military establishment heretofore trained and equipped by the Soviet Union - viewed alongside the huge, far-reaching aid program - dramatizes how closely and quickly Egypt and the United States have drawn together.

The new relationship, aside from its sheer size, raises a number of questions in a nation where leadership is largely based on Sadat's personality and which is surrounded by other Arab states hostile to his peace policies.

Because of their hostility, the growing U.S. involvement here has a number of implications for American relations with the Arab world as a whole. In the meantime, U.S. aid efforts here are closely identified with the United S tes and with Sadat's peace moves. Often, with the Arab tendency to view problems in terms of personalties, they are identified with President Carter.

An Egyptian driver capulized the developments of the past few years with Cairo humor when, after an American-provided bus roared away from a red light and covered him with exhaust fumes, he said: "that's the Voice of America."

He was smiling as he said it.

In general, U.S. involvement seems to be viewed as a good thing by most Egyptians who do not share the worries of some intellectuals. The aid fits into the widespread impression among Egyptian people that Sadat's peace policies will bring them prosperity or at least reduce their poverty.

The question arises how the "Voice of America" would sound if those policies went sour or if, after a few years of peace, Egyptians found their country still could not offer them a significantly better life.