Maxwell's Equations in Integral Form

Maxwell's equations are the four fundamental laws of electrodynamics, playing a role for electromagnetism analogous to Newton's laws in mechanics: together they completely describe the behaviour of electric charges, currents, and electric and magnetic fields. Each can be written either in integral form (as here, since the differential form is beyond higher-secondary level) or in an equivalent differential form. The first equation is Gauss's law for electricity, $\displaystyle\oint\vec E\cdot d\vec A=\dfrac{Q_{enclosed}}{\epsilon_0}$, which relates the net electric flux through any closed surface to the net charge enclosed by it; it holds for both discrete and continuous charge distributions, and implies that electric field lines begin on positive charges and end on negative charges, so an isolated positive or negative charge can genuinely exist on its own. The second equation is Gauss's law for magnetism, $\displaystyle\oint\vec B\cdot d\vec A=0$, stating that the net magnetic flux through any closed surface is always exactly zero; because this is true for every possible closed surface, magnetic field lines must always form continuous closed loops with no beginning or end, which is the mathematical statement that no isolated magnetic monopole (an isolated north or south magnetic charge) has ever been observed to exist. The third equation is Faraday's law of electromagnetic induction, $\displaystyle\oint\vec E\cdot d\vec l=-\dfrac{d\Phi_B}{dt}$, stating that the line integral of the electric field around any closed path equals the negative rate of change of magnetic flux through the surface bounded by that path; this is the law underlying every modern electrical generator and transformer. The fourth equation is the modified (Ampere-Maxwell) form of Ampere's circuital law, $\displaystyle\oint\vec B\cdot d\vec l=\mu_0 i_C+\mu_0\epsilon_0\dfrac{d\Phi_E}{dt}$, stating that the line integral of the magnetic field around any closed path is produced jointly by the conduction current and the displacement current (Maxwell's correction term) passing through the surface bo …