To determine which of the given Maxwell's equations is valid for time-varying conditions but not valid for static conditions, we need to examine each equation and its applicability:
\( ∮\vec{B}.\vec{dl}=μ _0I \):
This is Ampère's Circuital Law, which relates the magnetic field around a closed loop to the current passing through the loop.
In its original form, without displacement current, it is valid for static conditions (constant currents) but needs modification for time-varying fields by introducing the displacement current term.
This is Faraday's Law of Electromagnetic Induction which indicates that a time-varying magnetic field induces an electromotive force (EMF).
It is explicitly valid for time-varying conditions because it involves the rate of change of magnetic flux (\(\frac{\partialϕ_B}{\partial t}\)).
For static conditions (constant magnetic field), this term becomes zero (\(\frac{\partialϕ_B}{\partial t} = 0\)), thus making the equation not applicable.
Hence, this is the correct option.
\( ∮\vec{D}.\vec{dA}=Q \):
This is Gauss's Law for Electric Fields, stating that the electric flux through a closed surface is proportional to the charge enclosed.
This law is valid in both static and time-varying conditions because it is independent of time variation and depends only on the charge enclosed.
\( ∮\vec{E}.\vec{dl}=0 \):
This represents the static condition of no time-varying magnetic field, implying no induced EMF.
This condition is true only for electrostatic scenarios and is not valid under time-varying fields, making it not applicable to time-varying circumstances.
Therefore, the equation \( ∮\vec{E}.\vec{dl}=-\frac{\partialϕ_B}{\partial t} \) is specifically valid for time-varying conditions (Faraday's Law of Induction), highlighting the induced EMF due to changing magnetic fields, thus making it invalid for purely static conditions.
