Write the mathmatical equation of Ampere-maxwell law.

Options

A. `ointE.dS = (q)/(epsi_(0))`
B. `ointE.dl = - (dphi_(E))/(dt)`
C. `ointB.dS = 0`
D. `ointB.dl = mu_(0)(l_(c) + epsi_(0)(dphi_(E))/(dt))`

Explanation

To write the mathematical equation of the Ampere-Maxwell law, we start with the original form of Ampere's law and then incorporate Maxwell's modification.

### Step-by-Step Solution:

1. **Start with Ampere's Law**:
The original form of Ampere's law states that the line integral of the magnetic field \( \mathbf{B} \) around a closed loop is proportional to the total current \( I_{\text{inc}} \) passing through that loop.
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{inc}} \]
where \( \mu_0 \) is the permeability of free space.

2. **Introduce Displacement Current**:
Maxwell recognized that a changing electric field can also produce a magnetic field. He introduced the concept of displacement current \( I_D \) to account for this effect. The displacement current is given by:
\[ I_D = \epsilon_0 \frac{d\Phi_E}{dt} \]
where \( \epsilon_0 \) is the permittivity of free space and \( \Phi_E \) is the electric flux.

3. **Combine Currents**:
The total current \( I_{\text{total}} \) that contributes to the magnetic field is the sum of the conduction current \( I_C \) and the displacement current \( I_D \):
\[ I_{\text{total}} = I_C + I_D \]

4. **Write the Ampere-Maxwell Law**:
Substituting \( I_D \) into the original Ampere's law gives us the Ampere-Maxwell law:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( I_C + \epsilon_0 \frac{d\Phi_E}{dt} \right) \]

5. **Final Form**:
The complete mathematical expression for the Ampere-Maxwell law is:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_C + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} \]