in my revision guide it gives the equation for Faraday's law as $$\text{induced emf} = N\frac{\Delta \phi}{\Delta t}$$ while the one for Lenz's law is given $$\text{induced emf}=-N\frac{\Delta\phi}{\Delta t}$$. Here $N$ is the number of turns on the coil in question. My question is: Shouldn't the first equation say "magnitude of induced emf"? And second: Can't we just use the second equation in all instances when working out induced emf? (In other words there should only be one equation)
In a differential vector-calculus form the law can be written $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \,,$$ and here the minus sign is algebraically correct.