A federal judge in Alexandria has awarded $12,500 in damages to a former Alexandria woman whose home was broken into two years ago by FBI agents and Alexandria police who mistakenly thought fugitive Patty Hearst was staying there.

Judge Robert R. Merhige ordered the federal government to pay the money to Elizabeth Ann Norton of Wheeling, W. Va., after two days of testimony last week to determine how much Miss Norton should receive. Merhige ruled in January that the officers had acted illegally and that the government was liable for damages. Assistant U.S. Attorney James R. Hubard said the government will appeal the January ruling.

Miss Norton sued the government for $350,000 because she said the law enforcement officers violated her rights to privacy and that they forcibly entered her apartment without a search warrant.

Alexandria police said they had received an anonymous tip at 8:11 p.m. on March 15, 1975, that Hearst was living in Apartment 10 at 649 Notabene Dr., according to facts agreed to by both Miss Norton and the government.

"You know Patty Hearst is supposed to be in Pennsylvania," police said the caller told them. "But she's not. She's currently at 649 Notabene Dr., Apartment 10 in Alexandria and she's been there for the last week or 10 days. She has cut her hair and she's with one of the people who left with her from California."

Police later said the call probably came from a neighbor with whom Miss Norton often quarreled, the records showed.

Shortly after 10 p.m. two FBI agents and two Alexandria detectives knocked on Miss Norton's door and said they were law officers. She asked for identifications. When they did not produce any, she started to call the FBI, according to the records.

The men began pushing the door and hitting it, according to the records. Miss Norton eventually opened the door and the officers entered with weapons drawn. They searched the apartment and found no trace of Miss Hearst or her companions, the records showed.

The government contended that the officers had acted with "care, caution and clear respect for" Miss Norton's dignity.

Miss Norton testified last week that she was extremely upset about the incident and that she had to have psychiatirc care to relieve her nervousness as a result of the incident.