Which of the following Maxwell's equation is valid for time varying conditions but not valid for static conditions:

We need to identify which of Maxwell's equations is valid for time-varying conditions but not for static conditions.

Analysis of each option:

Option A: $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I$$

This is Ampere's circuital law in its original (static) form. It is valid for magnetostatics but is incomplete for time-varying fields, where Maxwell's correction (displacement current) must be added: $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$$. So this equation is valid for static conditions only.

Option B: $$\oint \vec{E} \cdot d\vec{l} = 0$$

This states that the electric field is conservative, which holds only in electrostatics (no time-varying magnetic fields). For time-varying conditions, this equation does not hold.

Option C: $$\oint \vec{D} \cdot d\vec{A} = Q$$

This is Gauss's law for electricity. It is valid for both static and time-varying conditions.

Option D: $$\oint \vec{E} \cdot d\vec{l} = -\frac{\partial \Phi_B}{\partial t}$$

This is Faraday's law of electromagnetic induction. It describes how a changing magnetic flux induces an electromotive force. In static conditions, $$\frac{\partial \Phi_B}{\partial t} = 0$$, and the equation reduces to $$\oint \vec{E} \cdot d\vec{l} = 0$$, which is simply the electrostatic result. The non-trivial form with a non-zero time derivative is specifically meaningful only when fields are time-varying.

Among the given options, Option D (Faraday's law) is the equation that is specifically valid for time-varying conditions and becomes trivial for static conditions.

The correct answer is Option D.