## The 4 Maxwell's Equations: Important Basics You Need to Know

Simple explanation of the Maxwell's equations for beginners. The divergence integral and the curl integral theorems are also explained.

Maxwell equations and electromagnetic waves

Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

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Lesson ## The 4 Maxwell's Equations: Important Basics You Need to Know

Simple explanation of the Maxwell's equations for beginners. The divergence integral and the curl integral theorems are also explained.

## Questions & Answers

## Derivations & Experiments

Derivation ## Displacement Current for the 4th Maxwell Equation

Derivation of the displacement current as a correction to the fourth Maxwell equation. For this purpose we use a plate capacitor.

Derivation ## Coulomb's Law using 1st Maxwell Equation

Here you will learn how to derive Coulomb's law for two point charges from divergence theorem and Maxwell's first equation.

## Related formulas

Formula ## 1. Maxwell Equation in Integral Form (E-Field, Charge)

`$$ \oint_A \class{blue}{\boldsymbol{E}} ~\cdot~ \text{d}\boldsymbol{a} ~=~ \frac{\class{red}{Q}}{\varepsilon_0} $$`Formula ## 1. Maxwell Equation (Differential Form)

`$$ \nabla ~\cdot~ \class{blue}{\boldsymbol{E}} ~=~ \frac{\class{red}{\rho}}{\varepsilon_0} $$`Formula ## 2. Maxwell equation (differential form)

`$$ \nabla \cdot \class{violet}{\boldsymbol{B}} ~=~ 0 $$`Formula ## 2. Maxwell equation (integral form)

`$$ \oint_A \class{violet}{\boldsymbol{B}} ~\cdot~ \text{d}\boldsymbol{a} ~=~ 0 $$`Formula ## 3. Maxwell equation (differential form)

`$$ \nabla \times \class{blue}{\boldsymbol{E}} ~=~ -\frac{\partial \class{violet}{\boldsymbol{B}}}{\partial t} $$`Formula ## 3. Maxwell equation (integral form)

`$$ \oint_{L} \class{blue}{\boldsymbol{E}} ~\cdot~ \text{d}\boldsymbol{l} ~=~ -\int_{A} \frac{\partial \class{violet}{\boldsymbol{B}} }{\partial t} ~\cdot~ \text{d}\boldsymbol{a} $$`Formula ## 4. Maxwell Equation of Magnetostatics (Differential Form)

`$$ \nabla \times \class{violet}{\boldsymbol{B}} ~=~ \mu_0 \, \class{red}{\boldsymbol{j}} $$`Formula ## 4th Maxwell Equation in Integral Form

`$$ \oint_{L} \class{violet}{\boldsymbol{B}} ~\cdot~ \text{d}\boldsymbol{l} ~=~ \mu_0 \, \class{red}{I} ~+~ \mu_0 \, \varepsilon_0 \, \int_{A} \frac{\partial \class{blue}{\boldsymbol{E}}}{\partial t} ~\cdot~ \text{d}\boldsymbol{a} $$`Formula ## 4. Maxwell Equation of Electrostatics (Integral Form)

`$$ \oint_{L} \class{violet}{\boldsymbol{B}} ~\cdot~ \text{d}\boldsymbol{l} ~=~ \mu_0 \, \class{red}{I} $$`Formula ## Biot-Savart Law for a Thin Wire (Magnetic Field)

`$$ \class{violet}{\boldsymbol{B}}(\boldsymbol{r}) ~=~ \frac{\mu_0 \, I}{4\pi} \int_{S} \frac{\boldsymbol{r}-\boldsymbol{R}}{|\boldsymbol{r}-\boldsymbol{R}|^3} \times \text{d}\boldsymbol{s} $$`## Related Illustrations

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Lesson ## Electromagnetic wave and its E-field and B-field components

Here you will learnthe wave equations for the E-field and B-field of an electromagnetic wave and how it can be simplified to a plane wave.

## Derivations & Experiments

Derivation ## Wave Equation for E-field and B-field

Derivation of the wave equation for the electric and mangetic field from the decoupled Maxwell equations in vacuum.

Derivation ## Energy of the Electric Field

Derivation of the energy of the electric field (E-field) using the charging process of a sphere (and a plate capacitor).

Derivation ## Energy of the magnetic field

Derivation of the magnetic energy and energy density of the B-field using a current-carrying coil. The formulas are also valid in general.

## Related formulas

Formula ## Wave Equation for E-Field

`$$ \nabla^2 \, \class{gray}{\boldsymbol{E}} ~=~ \mu_0 \, \varepsilon_0 \, \frac{\partial^2 \class{gray}{\boldsymbol{E}}}{\partial t^2} $$`Formula ## Wave Equation for B-Field

`$$ \nabla^2 \, \class{violet}{\boldsymbol{B}} ~=~ \mu_0 \, \varepsilon_0 \, \frac{\partial^2 \class{violet}{\boldsymbol{B}}}{\partial t^2} $$`## Related Illustrations

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Derivation ## B-Field inside a Current-Carrying Coil

In this derivation, the magnetic field inside a long coil is calculated using Ampere's law when N, l, and I are given.

## Related formulas

Formula ## Coil (Inductance, Number of Turns)

`$$ L ~=~ \mu_0 \, \mu_{\text r} \, \frac{ A \, N^2 }{ l } $$`Formula ## Long Coil (Magnetic Field, Mumber of Turns, Current)

`$$ \class{violet}{B} ~=~ \mu_0 \, \mu_{\text r} \, \frac{ \class{red}{I} \, N }{ l } $$`Formula ## Magnetic Energy of a Coil

`$$ W_{\text m} ~=~ \frac{1}{2} \, L \, I^2 $$`Formula ## Induced Voltage (Inductance, Current)

`$$ U ~=~ - L \, \frac{\text{d} \class{red}{I}}{\text{d} t} $$`## Related Illustrations

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Derivation ## Magnetic Field Inside and Outside a Coaxial Cable

Derivation of the magnetic field (B-field) of a coaxial cable, inside and outside, by exploiting Ampere's law.

## Related Illustrations

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Derivation ## Magnetic Field of a Helmholtz Coil

Derivation of the homogeneous magnetic field in the center (on the symmetry axis) of the Helmholtz coil with radius R and distance d.

## Related formulas

Formula ## Helmholtz Coil (B-Field, Same Current Direction)

`$$ \class{violet}{B}(z) = \frac{\mu_0 \, I \, R^2 \, N}{2} \, \left[ \left( (z-d/2)^2 + R^2 \right)^{-3/2} + \left( (z+d/2)^2 + R^2 \right)^{-3/2} \right] $$`## Related Illustrations

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Lesson ## Linearly and Circularly Polarized Electromagnetic Waves

Here you will learn about polarized light, i.e. linearly or circularly polarized electromagnetic waves, and what characterizes them.

## Related Illustrations

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Lesson ## Magnetic Dipole in a External Magnetic Field

Magnetic dipole (as a conductor loop) is simply explained and what happens to it when the is placed in an external magnetic field.

## Derivations & Experiments

Derivation ## Magnetic Dipole - Torque, Energy and Force

Derivation of the potential energy (potential), torque and force on a magnetic dipole - expressed with the magnetic dipole moment.

## Related formulas

Formula ## Magnetic dipole moment

`$$ \class{red}{\boldsymbol{\mu}} ~=~ \class{red}{I} \, \boldsymbol{A} $$`Formula ## Magnetic Dipole (Potential Energy)

`$$ W_{\mu} = -\boldsymbol{\mu} \cdot \class{violet}{\boldsymbol{B}} $$`Formula ## Magnetic Dipole (Force)

`$$ \boldsymbol{F} ~=~ \nabla \left( \boldsymbol{\mu} \cdot \class{violet}{\boldsymbol{B}} \right) $$`## Related Illustrations

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Derivation ## Self-Inductance of Two Current-Carrying Wires

Derive the self-inductance and magnetic flux of two parallel current carrying conductors with opposite currents.

## Related formulas

Formula ## Inductance of Two Current-Carrying Wires

`$$ L ~=~ \frac{\mu_0 \, l}{\pi} \, \left[ \frac{1}{2} + \ln\left(\frac{d-R}{R}\right) \right] $$`## Related Illustrations