If in a region there is a time varying electrinc field then which of the following Maxwell equation will be most suitable ?

Options

A. `oint vec B. vec d=I epsi_(0)(dphi_(E))/(dt)`
B. `oint vec B. vec d=mu_(0)I`
C. `oint vec B. vec d=mu_(0)[I+epsi_(0)(dphi_(E))/(dt)]`
D. `oint vec B. vec d=mu_(0)[I+epsi_(0)(dphi_(B))/(dt)]`

Answer

To determine which Maxwell equation is most suitable for a region with a time-varying electric field, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Context**:
- We are given a region where there is a time-varying electric field. This implies that the electric flux in that region is changing over time.

2. **Identifying Relevant Maxwell's Equations**:
- Maxwell's equations consist of four fundamental equations that describe how electric and magnetic fields interact. The relevant equations for our scenario are:
- Gauss's Law for Electricity
- Gauss's Law for Magnetism
- Faraday's Law of Induction
- Ampere-Maxwell Law

3. **Analyzing the Time-Varying Electric Field**:
- A time-varying electric field can induce a magnetic field. This is described by the modified Ampere's Law, which includes the concept of displacement current.

4. **Modified Ampere's Law**:
- The original form of Ampere's Law states:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}} \]
- Maxwell modified this to include the displacement current (\(I_D\)) due to the changing electric field:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 (I_C + I_D) \]
- Here, \(I_C\) is the conduction current and \(I_D\) is the displacement current given by:
\[ I_D = \epsilon_0 \frac{d\Phi_E}{dt} \]
- Where \(\Phi_E\) is the electric flux.

5. **Conclusion**:
- Since we are dealing with a time-varying electric field, the most suitable Maxwell equation is the modified Ampere's Law, which accounts for both conduction and displacement currents. Thus, the correct answer is option number 3, which represents the modified Ampere-Maxwell Law.

### Final Answer:
The most suitable Maxwell equation for a region with a time-varying electric field is the modified Ampere-Maxwell Law.

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