The explosions occurred during a military operation in the town bordering the Palestinian Gaza Strip, witnesses said.
The military regularly conducts operations in Rafah to destroy smuggling tunnels to Gaza, amid a wider campaign to quell a militant Islamist insurgency in the northern Sinai peninsula.
The attack came a day after a suicide bomber killed four soldiers when he rammed his bomb-laden car into a checkpoint near the northern Sinai town of El-Arish.
Attacks on the army and police have increased since the army's July 3 ouster of Islamist president Mohamed Morsi and a subsequent crackdown on his supporters and members of his Muslim Brotherhood.
Militants have launched a series of brazen attacks this week, after at least 57 people were killed in clashes between security forces and Morsi supporters on Sunday, most of them in Cairo.
On Monday, three people were killed and around 50 wounded when a car bomb exploded outside a security building in Al-Tur, the capital of South Sinai.
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In the Suez Canal city of Ismailiya, gunmen killed six soldiers in an attack on an army patrol.
And in Cairo, unknown assailants fired rocket-propelled grenades at communication satellite dishes, damaging one.
Morsi, who ruled as Egypt's first democratically elected president for just one year, was overthrown by the military in July after massive street protests against him.
Since then, the army-installed authorities have launched a massive crackdown on his supporters.
Hundreds were killed when security forces dispersed pro-Morsi protest camps in August, and at least 2,000 Muslim Brotherhood members are behind bars.
The United States on Wednesday suspended deliveries of major military hardware and cash assistance to Egypt to signal deep concern over the mounting bloodshed and lack of a democratic transition.
But Washington will keep up assistance "to help secure Egypt's borders" and bolster "counterterrorism and proliferation, and ensure security in the Sinai," State Department spokeswoman Jen Psaki said.