Maxwell's Third Equation: A Derivation from Faraday's Law

Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields and their interactions with matter. These equations form the foundation of classical electromagnetism, uniting electricity, magnetism, and optics. Each of Maxwell's equations is a re-expression or a significant generalization of earlier empirical laws.

Understanding Maxwell's Equations

The four Maxwell's equations are:

• Gauss's Law for Electricity: This law relates the electric field to the distribution of electric charges. It states that the electric flux through any closed surface is proportional to the total electric charge enclosed within that surface.
• Gauss's Law for Magnetism: This law states that the net magnetic flux through any closed surface is always zero. This implies that there are no isolated magnetic poles (monopoles); magnetic fields always exist as dipoles (North and South poles).
• Faraday's Law of Induction (Maxwell's Third Equation): This law describes how a changing magnetic field induces an electromotive force (EMF) and thus an electric field. This is the core of electromagnetic induction.
• Ampere's Circuital Law with Maxwell's Correction: This law relates the magnetic field to electric currents and changing electric fields (displacement current). Maxwell added the displacement current term, which was crucial for predicting electromagnetic waves.

Faraday's Law of Electromagnetic Induction

Faraday's law of electromagnetic induction states that a changing magnetic flux through a coil or circuit induces an electromotive force (EMF) in that coil or circuit. This induced EMF then drives an induced current if the circuit is closed. Mathematically, it can be expressed as:

The induced EMF ($\mathcal{E}$) is given by:

$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

where $\Phi_B$ is the magnetic flux and $\frac{d\Phi_B}{dt}$ is the rate of change of magnetic flux with respect to time. The negative sign indicates Lenz's Law, meaning the induced EMF opposes the change in magnetic flux that produced it.

We know that EMF is also defined as the line integral of the electric field $\vec{E}$ around a closed loop, and magnetic flux $\Phi_B$ is the surface integral of the magnetic field $\vec{B}$ over a surface $A$. So, Faraday's law can also be written as:

$$\oint \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \int \vec{B} \cdot d\vec{A}$$

Derivation of Maxwell's Third Equation from Faraday's Law

Maxwell's third equation is essentially the differential form of Faraday's law of electromagnetic induction. To derive it, we apply Stokes' theorem to the integral form of Faraday's law.

Stokes' theorem states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of that vector field over any surface bounded by the loop:

$$\oint \vec{E} \cdot d\vec{l} = \int (\nabla \times \vec{E}) \cdot d\vec{A}$$

Substituting this into the integral form of Faraday's law:

$$\int (\nabla \times \vec{E}) \cdot d\vec{A} = -\frac{d}{dt} \int \vec{B} \cdot d\vec{A}$$

Assuming the surface of integration does not change with time, we can move the time derivative inside the integral on the right side:

$$\int (\nabla \times \vec{E}) \cdot d\vec{A} = -\int \frac{\partial \vec{B}}{\partial t} \cdot d\vec{A}$$

Since this equality must hold for any arbitrary surface $A$, the integrands themselves must be equal:

$$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$

This is Maxwell's third equation. It expresses that a time-varying magnetic field ($\frac{\partial \vec{B}}{\partial t}$) generates a circulating electric field ($\nabla \times \vec{E}$). This direct relationship is a fundamental aspect of electromagnetic waves.

Why Other Laws Are Not the Source

• Ampere's Circuital Law: This law (with Maxwell's correction) describes how electric currents and changing electric fields create magnetic fields. It's distinct from how changing magnetic fields create electric fields.
• Gauss's Law of Electrostatics: This law relates static electric fields to electric charges. It does not deal with time-varying fields or induction.
• Gauss's Law of Magnetostatics: This law states that there are no magnetic monopoles and that magnetic field lines are always closed loops. It concerns the nature of magnetic fields but not their generation by changing electric fields or vice versa.

Therefore, Maxwell's third equation is directly derived from and represents Faraday's law of electromagnetic induction, which describes the phenomenon of electromagnetic induction where changing magnetic fields produce electric fields.