Course Fundamentals of Electrodynamics
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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Questions & Answers
Derivations & Experiments
Related formulas
Formula $$ \oint_A \class{blue}{\boldsymbol{E}} ~\cdot~ \text{d}\boldsymbol{a} ~=~ \frac{\class{red}{Q}}{\varepsilon_0} $$1. Maxwell Equation in Integral Form (E-Field, Charge)
Formula $$ \nabla ~\cdot~ \class{blue}{\boldsymbol{E}} ~=~ \frac{\class{red}{\rho}}{\varepsilon_0} $$1. Maxwell Equation (Differential Form)
Formula $$ \nabla \cdot \class{violet}{\boldsymbol{B}} ~=~ 0 $$2. Maxwell equation (differential form)
Formula $$ \oint_A \class{violet}{\boldsymbol{B}} ~\cdot~ \text{d}\boldsymbol{a} ~=~ 0 $$2. Maxwell equation (integral form)
Formula $$ \nabla \times \class{blue}{\boldsymbol{E}} ~=~ -\frac{\partial \class{violet}{\boldsymbol{B}}}{\partial t} $$3. Maxwell equation (differential form)
Formula $$ \oint_{L} \class{blue}{\boldsymbol{E}} ~\cdot~ \text{d}\boldsymbol{l} ~=~ -\int_{A} \frac{\partial \class{violet}{\boldsymbol{B}} }{\partial t} ~\cdot~ \text{d}\boldsymbol{a} $$3. Maxwell equation (integral form)
Formula $$ \nabla \times \class{violet}{\boldsymbol{B}} ~=~ \mu_0 \, \class{red}{\boldsymbol{j}} $$4. Maxwell Equation of Magnetostatics (Differential Form)
Formula $$ \oint_{L} \class{violet}{\boldsymbol{B}} ~\cdot~ \text{d}\boldsymbol{l} ~=~ \mu_0 \, \class{red}{I} ~+~ \mu_0 \, \varepsilon_0 \, \int_{A} \frac{\partial \class{blue}{\boldsymbol{E}}}{\partial t} ~\cdot~ \text{d}\boldsymbol{a} $$4th Maxwell Equation in Integral Form
Formula $$ \oint_{L} \class{violet}{\boldsymbol{B}} ~\cdot~ \text{d}\boldsymbol{l} ~=~ \mu_0 \, \class{red}{I} $$4. Maxwell Equation of Electrostatics (Integral Form)
Formula $$ \class{violet}{\boldsymbol{B}}(\boldsymbol{r}) ~=~ \frac{\mu_0 \, I}{4\pi} \int_{S} \frac{\boldsymbol{r}-\boldsymbol{R}}{|\boldsymbol{r}-\boldsymbol{R}|^3} \times \text{d}\boldsymbol{s} $$Biot-Savart Law for a Thin Wire (Magnetic Field)
Related Illustrations
Electric flux and enclosed charge (1st Maxwell equation) Second Maxwell equation in integral form Imaginary Sphere Surface around a Point Charge E-field change generates B-field Fourth Maxwell equation: Constant electric current generates B-field Current + time-dependent E-field generate a rotating B-field and vice versa Current Generates Rotating Magnetic Field Current Depends on the Choice of the Enclosing Loop in the Plate Capacitor - 2
Derivations & Experiments
Related formulas
Formula $$ \nabla^2 \, \class{gray}{\boldsymbol{E}} ~=~ \mu_0 \, \varepsilon_0 \, \frac{\partial^2 \class{gray}{\boldsymbol{E}}}{\partial t^2} $$Wave Equation for E-Field
Formula $$ \nabla^2 \, \class{violet}{\boldsymbol{B}} ~=~ \mu_0 \, \varepsilon_0 \, \frac{\partial^2 \class{violet}{\boldsymbol{B}}}{\partial t^2} $$Wave Equation for B-Field
Related Illustrations
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Related formulas
Formula $$ L ~=~ \mu_0 \, \mu_{\text r} \, \frac{ A \, N^2 }{ l } $$Coil (Inductance, Number of Turns)
Formula $$ \class{violet}{B} ~=~ \mu_0 \, \mu_{\text r} \, \frac{ \class{red}{I} \, N }{ l } $$Long Coil (Magnetic Field, Mumber of Turns, Current)
Formula $$ W_{\text m} ~=~ \frac{1}{2} \, L \, I^2 $$Magnetic Energy of a Coil
Formula $$ U ~=~ - L \, \frac{\text{d} \class{red}{I}}{\text{d} t} $$Induced Voltage (Inductance, Current)
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Related Illustrations
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Related formulas
Formula $$ \class{violet}{B}(z) = \frac{\mu_0 \, I \, R^2 \, N}{2} \, \left[ \left( (z-d/2)^2 + R^2 \right)^{-3/2} + \left( (z+d/2)^2 + R^2 \right)^{-3/2} \right] $$Helmholtz Coil (B-Field, Same Current Direction)
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Related Illustrations
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Derivations & Experiments
Related formulas
Formula $$ \class{red}{\boldsymbol{\mu}} ~=~ \class{red}{I} \, \boldsymbol{A} $$Magnetic dipole moment
Formula $$ W_{\mu} = -\boldsymbol{\mu} \cdot \class{violet}{\boldsymbol{B}} $$Magnetic Dipole (Potential Energy)
Formula $$ \boldsymbol{F} ~=~ \nabla \left( \boldsymbol{\mu} \cdot \class{violet}{\boldsymbol{B}} \right) $$Magnetic Dipole (Force)
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Related formulas
Formula $$ L ~=~ \frac{\mu_0 \, l}{\pi} \, \left[ \frac{1}{2} + \ln\left(\frac{d-R}{R}\right) \right] $$Inductance of Two Current-Carrying Wires
Related Illustrations
