Ampere's Law and Maxwell's Fourth Equation

Maxwell's equations are a set of four partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. These equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. One of these fundamental equations is directly based on Ampere's circuit law, with a crucial addition by Maxwell himself.

Maxwell's Equations Overview

Let's briefly look at all four of Maxwell's equations and their origins:

• Maxwell's First Equation: Gauss's Law for Electricity
  • This equation describes how electric charges produce an electric field. It states that the total electric flux through any closed surface is proportional to the total electric charge enclosed within that surface.
  • Differential Form: $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$
  • Integral Form: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$
• Maxwell's Second Equation: Gauss's Law for Magnetism
  • This equation states that there are no magnetic monopoles. It implies that magnetic field lines are continuous loops, always forming closed paths. The net magnetic flux through any closed surface is always zero.
  • Differential Form: $\nabla \cdot \vec{B} = 0$
  • Integral Form: $\oint \vec{B} \cdot d\vec{A} = 0$
• Maxwell's Third Equation: Faraday's Law of Induction
  • This equation describes how a time-varying magnetic field produces an electric field. It is the principle behind many electrical generators and transformers.
  • Differential Form: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$
  • Integral Form: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$
• Maxwell's Fourth Equation: Ampere-Maxwell's Law
  • This equation describes how both electric currents and changing electric fields produce magnetic fields. It is a modification of Ampere's original circuital law.
  • Ampere's original law stated that a magnetic field is produced by an electric current. However, Maxwell noticed that this law was incomplete when applied to time-varying fields, especially when considering the charging or discharging of a capacitor. He added a term called the "displacement current" ($\epsilon_0 \frac{\partial \vec{E}}{\partial t}$) to account for the changing electric flux, making the equation consistent with the continuity equation for charge.
  • Differential Form: $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$
  • Integral Form: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$

Fourth Equation and Ampere's Circuit Law

The question specifically asks which equation is based on Ampere's circuit law. As detailed above, Maxwell's fourth equation is the one that originates from Ampere's original circuit law but was extended by Maxwell to include the concept of displacement current. This extension was crucial for the theoretical prediction of electromagnetic waves.

Therefore, Maxwell's fourth equation is the correct answer as it is the fundamental relationship that evolved from Ampere's circuit law to form a complete and consistent theory of electromagnetism.