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

Options

A. `ointvecD*vec(dA)=Q`
B. `ointvecB*vec(dl)=mu_(0)I`
C. `ointvecE*vec(dl)=(delphi_B)/(delt)`
D. `ointvecB*vec(dl)=0`

Answer

To solve the question regarding which of Maxwell's equations is valid for time-varying conditions but not for static conditions, we can analyze the four Maxwell's equations in the context of static and dynamic fields.

### Step-by-Step Solution:

1. **Understand Maxwell's Equations**:
Maxwell's equations consist of four fundamental equations that describe how electric and magnetic fields interact. They are:
- Gauss's Law for Electricity: \(\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}\)
- Gauss's Law for Magnetism: \(\nabla \cdot \mathbf{B} = 0\)
- Faraday's Law of Induction: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\)
- Ampère-Maxwell Law: \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\)

2. **Identify Static and Time-Varying Conditions**:
- **Static Conditions**: These refer to situations where fields do not change with time. In static conditions, the partial derivatives with respect to time are zero (\(\frac{\partial \mathbf{E}}{\partial t} = 0\) and \(\frac{\partial \mathbf{B}}{\partial t} = 0\)).
- **Time-Varying Conditions**: These refer to situations where fields change over time, meaning the partial derivatives are not zero.

3. **Analyze Each Equation**:
- **Gauss's Law for Electricity**: Valid under both static and dynamic conditions.
- **Gauss's Law for Magnetism**: Valid under both static and dynamic conditions.
- **Faraday's Law of Induction**: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\) is valid only in time-varying conditions (not valid in static conditions since \(\frac{\partial \mathbf{B}}{\partial t} = 0\)).
- **Ampère-Maxwell Law**: \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) is also valid only in time-varying conditions (not valid in static conditions since \(\frac{\partial \mathbf{E}}{\partial t} = 0\)).

4. **Conclusion**:
The equations that are valid for time-varying conditions but not valid for static conditions are Faraday's Law of Induction and the Ampère-Maxwell Law. However, the question asks for one specific equation. Since Faraday's Law is often the most recognized in this context, we can conclude that:

**Final Answer**: The correct option is **Faraday's Law of Induction**.