AcademyElectromagnetic Induction
Academy
Maxwell's Equations
Level 1 - Physics topic page in Electromagnetic Induction.
Principle
Maxwell's equations connect electric and magnetic fields to charge, current, and changing flux.
Notation
\(\vec E\)
electric field
\(\mathrm{V\,m^{-1}}\)
\(\vec B\)
magnetic field
\(\mathrm{T}\)
\(q_{\mathrm{enc}}\)
enclosed charge
\(\mathrm{C}\)
\(I_{\mathrm{enc}}\)
enclosed conduction current
\(\mathrm{A}\)
\(\Phi_E\)
electric flux
\(\mathrm{V\,m}\)
\(\Phi_B\)
magnetic flux
\(\mathrm{Wb}\)
Method
Derivation 1: Sources of electric field
Gauss's law says electric flux through a closed surface measures enclosed charge.
Electric Gauss law
\[\oint\vec E\cdot d\vec A=\frac{q_{\mathrm{enc}}}{\epsilon_0}\]
Derivation 2: No magnetic monopoles
Magnetic field lines have no starts or ends, so net magnetic flux through a closed surface is zero.
Magnetic Gauss law
\[\oint\vec B\cdot d\vec A=0\]
Derivation 3: Fields induce fields
Changing magnetic flux creates circulating electric field; current and changing electric flux create circulating magnetic field.
Faraday law
\[\oint\vec E\cdot d\vec\ell=-\frac{d\Phi_B}{dt}\]
Ampere-Maxwell law
\[\oint\vec B\cdot d\vec\ell=\mu_0I_{\mathrm{enc}}+\mu_0\epsilon_0\frac{d\Phi_E}{dt}\]
Rules
These are Maxwell's equations in integral form.
Gauss electric
\[\oint\vec E\cdot d\vec A=\frac{q_{\mathrm{enc}}}{\epsilon_0}\]
Gauss magnetic
\[\oint\vec B\cdot d\vec A=0\]
Faraday law
\[\oint\vec E\cdot d\vec\ell=-\frac{d\Phi_B}{dt}\]
Ampere-Maxwell
\[\oint\vec B\cdot d\vec\ell=\mu_0I_{\mathrm{enc}}+\mu_0\epsilon_0\frac{d\Phi_E}{dt}\]
Wave speed
\[c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\]
Examples
Question
Which Maxwell equation says there are no isolated magnetic charges?
Answer
Gauss's law for magnetism:
\[\oint\vec B\cdot d\vec A=0\]
Checks
- Closed surface integrals use \(d\\vec A\); closed path integrals use \(d\\vec\\ell\).
- The minus sign belongs to Faraday's law.
- Displacement current is the changing electric flux term in Ampere-Maxwell law.
- In vacuum, the constants imply electromagnetic waves travel at \(c\).