For the first equation, the induced emf tends to oppose the flux changes AND the minus sign indicating the opposition BUT we often neglect the minus sign, seeking only magnitude of induce emf.
If we change the magnetic flux through a coil of N turns, an induced emf appears in every turn and the total induced in the coil is sum of these individual induced emfs. If the coil is tightly wound (closely packed), so that the same magnetic flux passes through all the turns, the total emf induced in the coil is shown as second equation.
To determine the direction of current using the "Right-Hand Rule":
Thumb- points in the direction of motion of the field relative to the conductor
Forefinger- points in the direction of that field
Middle finger- the direction of induced current
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