Bragg was the eldest child of Sir William Bragg. His maternal grandfather, Sir Charles Todd, was postmaster general and government astronomer of South Australia. Educated at St. Peter's College, Adelaide, and then at Adelaide University, Bragg gained high honours in mathematics at an age when most boys were still in secondary school.

In 1909 he went to England to enter Trinity College, Cambridge. He began the study of physics, which he had not studied earlier, although he had taken some chemistry. During the summer vacation of 1912, his father discussed with him a recent book on the work of the German physicist Max von Laue (Laue, Max von), who asserted that X rays could be diffracted by passing them through crystals. Upon his return to Cambridge, young Bragg, believing that Laue's explanation was incorrect in detail, carried out a series of ingenious original experiments, as the result of which he published the Bragg equation (Bragg law), which tells at what angles X rays will be most efficiently diffracted by a crystal when the X-ray wavelength and distance between the crystal atoms are known (see Bragg law). This equation is basic to X-ray diffraction, a process used to analyze crystal structure by studying the characteristic patterns of X rays that deviate from their original paths because of the closely spaced atoms in the crystal. He also showed that in rock salt (halite) the two kinds of atoms, sodium and chlorine, are arranged alternately, so that atoms of the same element never touch each other. Meanwhile, his father had designed the X-ray spectrometer, a device to make exact measurements of X-ray wavelengths. The two scientists spent vacations using the Bragg spectrometer to determine many other atomic arrangements, including that of diamond.

In 1914 Bragg became a fellow and lecturer in natural sciences at Trinity College. Later that year he and his father were jointly awarded the Barnard Gold Medal of the U.S. Academy of Sciences, the first of many such honours and awards. From 1915 to 1919, in World War I, Bragg served as technical adviser on sound ranging (determining the distance of enemy artillery from the sound of their guns) in the map section of British army headquarters in France, and he was there in 1915 when the Nobel Prize for Physics was awarded jointly to his father and himself for demonstrating the use of X rays for revealing the structure of crystals.

After the war Bragg succeeded Ernest Rutherford as professor of physics at Victoria University of Manchester, and there he built his first research school, for the study of metals and alloys and silicates. His work on silicates transformed a chemical riddle into a system of simple and elegant architecture. In 1921 he married Alice Hopkinson, a doctor's daughter, by whom he had two sons and two daughters. His wife's charm and character greatly helped him throughout his professional career. In the same year he was elected a Fellow of the Royal Society.

From 1937 to 1938 Bragg was director of the National Physical Laboratory, but he was impatient with committee work. About this period of his life he often used to remark that he found the engagement book, the in-tray, and the list of matters requiring urgent attention to be the deadly enemies of scientific work.

So he gladly left pure administration to succeed Rutherford again, this time as Cavendish Professor of Experimental Physics at Cambridge. Here he founded a second flourishing research school to study metals and alloys, silicates, and proteins, but he was also deeply concerned that science students have time to enjoy a full education and come to understand something of the meaning and purpose of life.

In January 1954 Bragg became director of the Royal Institution, London, as his father had been before 1940. He introduced several successful innovations: year-round lectures for schoolchildren, illustrated by demonstrations requiring apparatus too large or too costly for school resources (some 20,000 children attended each year); courses for science teachers; and lectures for civil servants whose early training had not included science. Popular and successful as a lecturer, Bragg was also in great demand for radio and television appearances. At an age at which many scientists lose interest in research, he built up a third research team, some of whose members successfully tackled the structures of complex organic crystals. Bragg retired from active scientific work in 1965.