Question

Write Maxwell equations for free space.

Short answer

Maxwell's Equations in free space: \( abla \bullet \textbf{E} = 0 \), \( abla \bullet \textbf{B} = 0 \), \( abla \times \textbf{E} = -\frac{\text{d}\textbf{B}}{\text{d}t} \), \( abla \times \textbf{B} = \frac{1}{c^2} \frac{\text{d}\textbf{E}}{\text{d}t} \)

Step-by-step solution

Gauss's Law for Electricity

Gauss's Law for Electricity states that the electric flux through a closed surface is proportional to the charge enclosed. In free space, where there are no charges, this law can be written as: \[ abla \bullet \textbf{E} = 0 \]

Gauss's Law for Magnetism

Gauss's Law for Magnetism states that there are no magnetic monopoles, and hence the net magnetic flux through a closed surface is zero. This can be written as: \[ abla \bullet \textbf{B} = 0 \]

Faraday's Law of Induction

Faraday's Law of Induction states that a changing magnetic field induces an electric field. In the absence of charges, this law is: \[ abla \times \textbf{E} = -\frac{\text{d}\textbf{B}}{\text{d}t} \]

Ampere’s Law (with Maxwell’s correction)

Ampere’s Law states that magnetic fields can be generated by electric currents and by changing electric fields. In free space, this is: \[ abla \times \textbf{B} = \frac{1}{c^2} \frac{\text{d}\textbf{E}}{\text{d}t} \]

Key concepts

Gauss's Law for Electricity

Gauss's Law for Electricity explains how electric fields originate from charges. It tells us that the total electric flux through a closed surface is equal to the charge enclosed divided by the electric constant. In mathematical terms, this is written as: \[ abla \bullet \textbf{E} = \frac{\rho}{\boldsymbol{\text{ε}_0}} \]
However, in free space where there are no charges present, the equation simplifies to: \[ abla \bullet \textbf{E} = 0 \]
This implies that the electric flux through any closed surface in free space is zero, meaning no net electric charge is enclosed.

Gauss's Law for Magnetism

Gauss's Law for Magnetism states that there are no magnetic monopoles. It basically means that the magnetic field lines are always continuous loops and they never begin or end. Mathematically, it is represented as: \[ abla \bullet \textbf{B} = 0 \]
This indicates that the total magnetic flux through a closed surface is zero, reflecting that one cannot isolate a single magnetic pole or 'monopole' as we can with electric charge. Similar to how for any given point, the number of magnetic field lines entering equals the number leaving.

Faraday's Law of Induction

Faraday’s Law of Induction describes how a changing magnetic field creates (or induces) an electric field. This is a cornerstone of electromagnetism, underlying many technologies such as transformers and electric generators. It can be expressed mathematically as: \[ abla \times \textbf{E} = -\frac{\text{d}\textbf{B}}{\text{d}t} \]
This equation tells us that if there is a time-varying magnetic field at a certain location, it will result in a circulating electric field around that location. The minus sign indicates the direction of the induced electric field is such that it tends to oppose the change in the magnetic field (Lenz's Law).

Ampere's Law with Maxwell's Correction

Ampere’s Law, initially only considered electric currents as sources of magnetic fields. However, James Clerk Maxwell added a term to account for changing electric fields, leading to it’s modern form called Ampere's Law with Maxwell's Correction. It is written as: \[ abla \times \textbf{B} = \frac{1}{c^2} \frac{\text{d}\textbf{E}}{\text{d}t} \]
This equation means that both electric currents and changing electric fields generate magnetic fields. The term \[ \frac{1}{c^2} \frac{\text{d}\textbf{E}}{\text{d}t} \] is Maxwell's correction, where \[ c \] is the speed of light. Thanks to this correction, Maxwell unified electricity and magnetism, demonstrating that they are interrelated phenomena.