1. We check which equation is valid for time-varying fields but not for static fields.

2. Examine each option:

   A. $\oint \vec E\cdot d\vec l=0$

   • This is the electrostatic form of Faraday’s law.
   • It is valid for static conditions.
   • For time-varying magnetic field, the correct relation is $\oint \vec E\cdot d\vec l=-\frac{d\phi_B}{dt}$
   • So A is not the required equation.

   B. $\oint \vec B\cdot d\vec l=\mu_0 I$

   • This is Ampere’s circuital law for steady currents.
   • For time-varying conditions, Maxwell corrected it to $\oint \vec B\cdot d\vec l=\mu_0 I+\mu_0\epsilon_0\frac{d\phi_E}{dt}$
   • Hence the given form is mainly for static/steady case, not the required one.

   C. $\oint \vec E\cdot d\vec l=-\frac{\partial \phi_B}{\partial t}$

   • This is Faraday’s law of electromagnetic induction.
   • It applies when magnetic field changes with time.
   • In static conditions, $\frac{\partial \phi_B}{\partial t}=0$ and it reduces to $\oint \vec E\cdot d\vec l=0$
   • Thus this is the equation characterizing time-varying conditions, not static conditions in its general form.
   • So this is the correct option.

   D. $\oint \vec D\cdot d\vec A=Q$

   • This is Gauss’s law.
   • It is valid for both static and time-varying situations.
   • So D is not correct.

3. Therefore, the Maxwell equation valid for time-varying conditions but not for static conditions is:

$\boxed{\oint \vec E\cdot d\vec l=-\frac{\partial \phi_B}{\partial t}}$

So, the correct option is C.