JOHANNESBURG, South Africa (AP) _ Police today said they have detained a black Anglican bishop under a security law designed to urb civil unrest. Bishop Desmond Tutu said authorities should ''come to their senses'' and release his colleague.

Meanwhile, thousands of people took to the streets in Soweto today to protest the arrest of 15 youths on charges of killing a policeman. Officials said a policeman was fatally stabbed in today's unrest, and witnesses reported three protesters were wounded by police.

Tutu, the Anglican archbishop-elect of southern Africa, said police took Bishop Sigisbert Ndwandwe into custody Thursday night and were holding him on a charge of public violence.

Ndwandwe, 55, is suffragan bishop of the West Rand, the region of Transvaal province west of Johannesburg. A suffragan bishop in the Anglican Church is an assistant to a diocesan bishop.

On Wednesday, Ndwandwe's residence near the city of Klerksdorp, about 90 miles southwest of Johannesburg, was fire-bombed, but there was no immediate indication that the incident had any connection with his detention.

The bishop's daughter, Angela, said in a telephone inteview that her father, who was ordained in 1978, has worked closely with anti-apartheid activitists. She said the police officers who came to their home Thursday gave no explanation why the bishop was being detained.

Maj. Steve Von Rooyen, a spokesman at police headquarters in Pretoria, said Ndwandwe was being held under a section of the Internal Security Act that allows police to detain without a warrant people whose actions are judged to be contributing to unrest.

Von Rooyen said he was not authorized to release details about the circumstances of the bishop's detention. Angela Ndwandwe said her father was being held at a police station in Stilfontein. She said she had not heard from him personally, but understood he had been allowed to see a lawyer.

Tutu, the 1984 Nobel Peace Prize winner and one of South Africa's most prominent black leaders, said he was distressed by Ndwande's detention.

''If anyone has been working towards holding together a community that was exploding, then it was this person,'' Tutu said. ''This is not the way to go around defusing this explosive situation.''

''I can only hope that they will come to their senses and release him quickly.''

Disturbances were reported today throughout Soweto, a black township of about 2 million people outside Johannesburg.

Thousands of people, many of them teen-agers, took to the streets to protest the arrest of 15 youths on charges they burned to death a black policeman. Protesters, on foot and in vehicles, tried to reach the courthouse where the youths were appearing, but were stopped by police road blocks and tear gas.

Police said a black police sergeant was stabbed to death by a crowd of blacks. In another area of Soweto, witnesses said three people were wounded by police gunfire when officers tried to disperse youths stopping motorists to demand rides to the courthouse.

In the nothern town of Brits, several hundred white supporters of far-right groups disrupted a planned address by the government's deputy minister of information, Louis Nel, making noise for more than an hour and forcing Nel to leave without giving his speech.

''This is the ugly face of the Afrikaner - white radicalism at its worst,'' Nel told reporters.

Those making the disturbance were members or supporters of white- supremacist political or paramilitary groups who contend that the dominant National Party is making too many concessions to blacks.

At one point, demonstrators tore up the National Party flag, and others held aloft the banner of the Afrikaner Weerstandsbeweging, or Afrikaner Resistance, a far-right paramilitary group.

President P.W. Botha, the National Party's leader, said in a speech Thursday night in Vereeniging, south of Johannesburg, that the government ''will have to give political rights to all who do not have them.''

On Wednesday, Botha's government abandoned enforcement of pass law regulations which restricted movements of blacks. The pass laws were one of the pillars of apartheid, the racial segregation system that empowers 5 million whites to rule 24 million voteless blacks.

''I am not scared to sit around a table with blacks,'' Botha in his speech. ''I have done it often and I have not turned black, nor have they turned white.''

''Today we have arrived at the emancipation from guardianship of the black and the brown and the rejection of the colonial domination of the past,'' he said.