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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)

NB. emf = electromotive force

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    $\begingroup$ This equation is the integrated form of Maxwell equation dmckee posted. Thus the sign is only meaningful if you choose a correct sign convention. Suppose you have a circular wire. Should you take clockwise emf as positive or negative? Should you take upward flux change as positive or negative? The answer is you should use right hand rule. If counterclockwise emf is positive then upward flux change is positive and there should be a minus sign in the equation. $\endgroup$
    – Azad
    Commented May 5, 2015 at 19:26

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The one tells you only the magnitude of the induced field. The minus sign associated with Lens' Law is a reminder of a procedure for determining the directional sense of the induced EMF (tending to opposed the applied change in flux).

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.

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