How we get sense of direction from faraday's law while it as a whole is scalar?

Question

I'm referring to Maxwell's third equation
If we assume direction of existing magnetic field i.e. M.F due to magnet as positive then if I look at Right hand side of equation which is $frac{-d∅}{dt}$ then in direction sense it means that it will always give us the direction of induced emf or direction of induced current indirectly (if I take direction of induced emf as sense of rotation by right hand thumb rule).
Now if I consider Left hand side of equation also then I also have to take assumptions to slove left hand side for equation to hold true for induced emf like direction of dl is taken (by sense of right hand thumb rule) in the direction of existing magnetic field and if limits of integration goes from L1 to L2 then $L2>L1.$
Now if I consider negative sign on right hand side then it has a big role in defining the equation and for it to hold true considering all assumptions. Is that how negative sign is used and how it is related with Lenz law?
If we do not take negative sign on right hand side and take assumptions such that there is no need of negative sign for equation to hold true then we know the reality that induced emf is always in the direction in accordance with Lenz law. Now no negative sign is used then how can we say that negative sign signifies Lenz law as there is no negative sign and obviously reality is not changed equation always follows Lenz law.
I want to ask that are those assumptions responsible in defining what sign will signify?

J. Murray · Answer

When we compute the flux $phi_B=iint vec B cdot mathrm dvec S$, we need a surface $S$ which has an orientation, i.e. a choice of normal vector at each point.

The boundary of an oriented surface inherits an orientation via the right-hand rule - specifically, if your thumb points in the direction of $mathrm dvec S$, your fingers curl in the direction of $mathrm dvec ell$:

Faraday's law is often written in integral form as $mathcal E equiv oint vec E cdot mathrm dvec ell = -frac{d}{dt}phi_B$.  What this means is the following.  Consider any oriented surface you like, and compute the magnetic flux through that surface.  Next, compute the EMF $mathcal E = oint vec E cdot mathrm d vec ell$ around the boundary of that surface, which inherits its own orientation via the right hand rule.  We then have that $mathcal E = -frac{dphi_B}{dt}$.
Specifically, let's choose the orientation of the surface as drawn in my figures, and say that $frac{dphi}{dt}>0$.  The minus sign in Faraday's law means that the induced EMF is oriented opposite to the blue arrows, i.e. the closed loop integral is negative.

How we get sense of direction from faraday's law while it as a whole is scalar?

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