The rabbis, scientists and engineers of the Zomet Institute are trying to solve the problems that arise when technology and the Torah collide.

Working from their research facility in the West Bank settlement of Alon Shvut, they create electronic devices  from phones to alarm systems to motorized vehicles  that obey Orthodox Jewish laws about the Sabbath, when even turning an electric current on or off is forbidden.

“We’re trying to combine making a modern Jewish state with age-old Jewish law,” said Dan Marans, executive director of Zomet. That requires both a deep knowledge of Judaism’s legal code, or halacha, and a bit of ingenuity.

“Every day, God gives us things to take advantage of,” Mr. Marans said. “We just have to know how.”

For decades, research groups like Zomet enjoyed a near-monopoly on the kosher gadget industry. They sold most of their inventions to the Israeli government and military.