What is the name associated with the equation \(E^{\longrightarrow} \cdot d t^{-}=-(d \Phi \beta / d t)\) (A) Gauss law for electricity (B) Gauss law for magnetism (C) ampere's law (D) faraday's law

The given equation is: \(E^{\longrightarrow} \cdot dt^{-} = -(\frac{d\Phi \beta}{dt})\) where \(E^{\longrightarrow}\) represents the electric field, \(dt^{-}\) is a change in time, and \(d\Phi \beta / dt\) represents the rate of change of magnetic flux. This equation indicates that the electric field is related to the rate of change of magnetic flux.

Now, let's compare the given equation with the laws mentioned in the options: (A) Gauss's law for electricity: It deals with the electric flux through a closed surface and the enclosed electric charge, not the relation between the electric field and magnetic flux. (B) Gauss's law for magnetism: It states that the net magnetic flux through a closed surface is zero. This law is also not related to the given equation. (C) Ampere's law: It relates the circulation of the magnetic field around a closed loop to the net current passing through the loop. It also doesn't match with our equation. (D) Faraday's law: Faraday's law of electromagnetic induction states that the electromotive force (EMF) induced in a circuit is proportional to the rate of change of magnetic flux. It can be mathematically represented as: \(EMF = -\frac{d\Phi \beta}{dt}\) Comparing the given equation with Faraday's law, we can see a clear connection between them since both deal with the relationship between the electric field and the rate of change of magnetic flux.

Based on the comparison made in step 2, it is evident that the given equation is related to (D) Faraday's law.