Sinha's early education started in Lake School for girls, in Kolkata, which was founded by her elder sister Sushama Sengupta. She then attended Asutosh College, followed by the Scottish Church College, and finally the prestigious Rajabazar Science College, University of Calcutta. She became the first woman to earn a PhD in Physics from Calcutta University in 1956. She was mentored by Professor Satyendra Nath Bose, a theoretical physicist known for his work in quantum mechanics.