One policeman was slightly wounded when his patrol car was hit by two roadside bombs in a Cairo suburb, the interior ministry said.
The capital has been rocked by several bombings and shootings targeting police over the past week.
On January 24, a day before the third anniversary of the 2011 uprising that toppled Hosni Mubarak, a car bombing hit police headquarters, one of four bombs targeting the force in the city.
Four days later, police general Mohamed Saeed, an aide to the interior minister, was killed outside his home by gunmen on a motorbike.
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Supporters of Morsi, who was ousted by the army on July 3, had called for protests today.
Police fired tear gas to disperse a pro-Morsi demonstration when the deposed Islamist's supporters clashed with opponents in the Mediterranean city of Alexandria, security officials said.
Similar pro-Morsi demonstrations were staged in Cairo, Sharqiya and Fayoum, they said.
The military-installed authorities have launched a deadly crackdown on Morsi's supporters, mostly Islamists, since his removal.
Morsi supporters continue to stage regular protests demanding his reinstatement.