What is Maxwell displacement current? 

Answer

### Step-by-Step Solution:

1. **Understanding the Context**:
- The concept of Maxwell Displacement Current arises from the need to modify Ampere's Circuital Law to account for situations where electric fields change over time, particularly in the case of capacitors.

2. **Capacitor Charging Scenario**:
- When a capacitor is connected to a battery, it does not charge instantaneously. Instead, the charging process takes time, and during this period, the current flowing from the battery increases gradually.

3. **Electric Field Variation**:
- As the capacitor charges, the electric field between its plates also changes. This changing electric field is crucial because it indicates that we are not in a steady state.

4. **Magnetic Field Generation**:
- According to Ampere's Circuital Law, a magnetic field is generated by a current. However, in the case of a changing electric field, we need to consider that the electric field itself can also induce a magnetic field.

5. **Modification of Ampere's Law**:
- Maxwell recognized that the original Ampere's Law did not account for the changing electric field. Therefore, he introduced the concept of "displacement current" to address this gap.

6. **Displacement Current Definition**:
- The displacement current (\(I_d\)) is defined mathematically as:
\[ I_d = \epsilon_0 \frac{d\Phi_E}{dt} \]
where \(\epsilon_0\) is the permittivity of free space and \(\Phi_E\) is the electric flux.

7. **Maxwell-Ampere Circuital Law**:
- The modified form of Ampere's Law, which includes the displacement current, is given by:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 (I + I_d) \]
where \(I\) is the conduction current and \(I_d\) is the displacement current.

8. **Conclusion**:
- The displacement current is essential for explaining how changing electric fields can produce magnetic fields, thus completing the symmetry in Maxwell's equations and allowing for the propagation of electromagnetic waves.