Maxwell's Equations

Maxwell's set of four equations summarises all of classical electromagnetism, but Ampere's original circuital law, \oint \vec{B}\cdot d\vec{l} = \mu_0 I, was incomplete because it failed in regions like a capacitor gap where no conduction current flows. Maxwell repaired it by adding the displacement-current term, giving the Ampere-Maxwell law \oint \vec{B}\cdot d\vec{l} = \mu_0\left(I + \varepsilon_0\frac{d\Phi_E}{dt}\right). This addition expresses that a changing electric field generates a magnetic field, completing the symmetry with Faraday's law. Gauss's law for electricity relates flux to enclosed charge and needed no such term. Gauss's law for magnetism merely states that magnetic monopoles do not exist. Faraday's law already described how a changing magnetic field induces an electric field and was left unchanged. Recognising which law carries the displacement-current correction is explicitly tested in the NCERT and JEE syllabus. As a consistency check, only the amended Ampere law makes the four equations symmetric in the way the two fields generate each other, which is precisely what allows self-sustaining electromagnetic waves to exist.