The formula for induced e.m.f. in a coil due to change in magnetic flux through the coil is (here A = area of the coil, B = magnetic field)
e=−A⋅dBdt
e=−B⋅dAdt
e=−ddt(A⋅B)
e=−ddt(A×B)
e=-dϕdt=−ddt(A⋅B)