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Assuming there is a time varying magnetic field (B), how to determine the direction of the induced electric field due to B. and to which parameters does it depend?

P.S. determining the electric field in an arbitrary point in space such as P

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The induced electric field (by electromagnetic induction which is the name of the phenomenon you are asking about) is combined with a pre-existing electric field from other sources. It is not really possible to exactly divide which "part" of the electric field is due to the changes of the magnetic fields and which part is due to other causes.

Consequently, only the derivatives of the electric field are related to the changes of the magnetic field via the equation $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} $$ which is the Maxwell-Faraday equation, one of the well-known Maxwell's equations. A consequence of this equation (and of some related laws affecting the motion of charge carriers in conductors) is the Kelvin-Stokes theorem $$ \oint_{\partial \Sigma} \mathbf{E} \cdot d\boldsymbol{\ell} = - \int_{\Sigma} \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A} $$ which says that the voltage (more precisely EMF, the electromotive force) along a closed contour is the (minus) time derivative of the magnetic flux (integrated magnetic field over the area) through the area bordered by the closed contour. The proper relationship between $\boldsymbol{\ell}$ (fingers) and the direction of $\mathbf{A}$ (thumb) is given by the right hand rule.

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