Question

As indicated below, which one is the equation of Ampere-Maxwell law?

Hint:

The {Ampere-Maxwell Law} states that a {changing electric field produces a magnetic field}, forming the foundation of {electromagnetic waves}. The term \( \mu_0 \varepsilon_0 \frac{d\Phi_B}{dt} \) accounts for the displacement current.

Answer 1

Step 1: Understanding Ampere-Maxwell Law 
The Ampere-Maxwell Law is a modification of Ampere’s Circuital Law, which originally stated: \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 i_c \] where:
- \( \oint \mathbf{B} \cdot d\mathbf{l} \) represents the circulation of the magnetic field around a closed loop,
- \( i_c \) is the conduction current enclosed by the loop,
- \( \mu_0 \) is the permeability of free space. 
Step 2: Maxwell's Addition - Displacement Current 
James Clerk Maxwell introduced the concept of displacement current, which accounts for the changing electric field, leading to the corrected equation: \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 i_c + \mu_0 \varepsilon_0 \frac{d\Phi_B}{dt} \] where:
- \( \varepsilon_0 \) is the permittivity of free space,
- \( \frac{d\Phi_B}{dt} \) represents the rate of change of electric flux, contributing to displacement current. 
Step 3: Importance of Ampere-Maxwell Law 
- This equation bridges the gap between electric and magnetic fields.
- It is one of Maxwell's four equations, crucial for electromagnetism.
- Explains how a changing electric field produces a magnetic field, leading to the concept of electromagnetic waves.