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The Maxwell Equations are 4 fundamental equations that describe the relation between the quantities of electromagnetism.

Name Integral Version Differential Version
Gauss’s law: Ω𝐄dA=1ϵ0ΩρdV\mathop{\circ\!\!\!\iint}_{\partial \Omega} \boldsymbol E \cdot \,\text{d}A = \frac{1}{\epsilon_0} \iiint_{\Omega} \rho \,\text{d}V div𝐃=𝜚\text{div}\boldsymbol D = \varrho
Faraday’s law of induction A𝐄d𝐬=A𝐁td𝐀\oint_{\partial A} \boldsymbol E \cdot \,\text{d}\boldsymbol s = - \iint_{A} \frac{\partial \boldsymbol B}{\partial t} \cdot \,\text{d}\boldsymbol A rot𝐄+𝐁t=0\text{rot}\boldsymbol E + \frac{\partial \boldsymbol B}{\partial t} = 0
Gauss’s law for magnetism Ω𝐁dA=0\mathop{\circ\!\!\!\iint}_{\partial \Omega} \boldsymbol B \cdot \,\text{d}A = 0 div𝐁=0\text{div}\boldsymbol B = 0
Ampère’s law A𝐇d𝐬=A𝐣d𝐀+A𝐃td𝐀\oint_{\partial A} \boldsymbol H \cdot \,\text{d}\boldsymbol s = \iint_{A} \boldsymbol j \cdot \,\text{d}\boldsymbol A + \iint_{A} \frac{\partial \boldsymbol D}{\partial t} \cdot \,\text{d}\boldsymbol A rot𝐇=𝐣+𝐃t\text{rot}\boldsymbol H = \boldsymbol j + \frac{\partial \boldsymbol D}{\partial t}

Gauss’s law

Explanation: The differential form states that if there exists electric charge somewhere, then the divergence of DD at that point is nonzero, otherwise it is equal to zero. The integral form states that the amount of charge inside a volume VV of enclosed charge is equal to the total amount of Electric Flux DD exiting the surface SS.