The maxwell’s equation :
`oint vec(B).vec(dl) = mu_(0) (i+varepsilon_(0).(dphi_(E))/dt)` is a statement of -

Options

A. Faraday’s law of induction
B. Modified Ampere’s law
C. Gauss’s law of electricity
D. Gauss’s law of magnetism

Answer

To solve the question regarding the Maxwell's equation given as:

\[ \oint \vec{B} \cdot \vec{dl} = \mu_0 \left( i + \varepsilon_0 \frac{d\Phi_E}{dt} \right) \]

we need to identify which of the four Maxwell's equations this represents.

### Step-by-Step Solution:

1. **Understand the Equation**: The equation provided is an integral form of one of Maxwell's equations. It relates the magnetic field (\(\vec{B}\)) around a closed loop to the current (\(i\)) passing through that loop and the rate of change of electric flux (\(\frac{d\Phi_E}{dt}\)).

2. **Identify the Components**:
- The left side, \(\oint \vec{B} \cdot \vec{dl}\), represents the circulation of the magnetic field around a closed path.
- The right side consists of two terms: \(\mu_0 i\) (the contribution from the current) and \(\mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}\) (the contribution from the changing electric field, known as displacement current).

3. **Recall Maxwell's Equations**: The four Maxwell's equations are:
- Gauss's Law for Electricity
- Gauss's Law for Magnetism
- Faraday's Law of Induction
- Ampère-Maxwell Law (which includes the displacement current term)

4. **Match the Equation**: The equation given matches the Ampère-Maxwell Law, which is a modification of Ampère's circuital law to include the displacement current. The original form of Ampère's law only considered current, while the modified form includes the term for the changing electric field.

5. **Conclusion**: Since the equation provided is the modified form of Ampère's law, we conclude that it represents the **Ampère-Maxwell Law**.

### Final Answer:
The equation represents the **Modified Ampère's Law** (Option B).

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