Q.Write down the integral form of modified Ampere's circuital law.

The four fundamental laws of electrodynamics, playing the same organising role for electromagnetism that Newton's three laws play for mechanics -- together they completely determine the behaviour of electric charges, currents, and electric and magnetic fields, and can be written in either integral form (used at this level) or an equivalent differential form. First equation -- Gauss's law for electricity: $\displaystyle\oint\vec E\cdot d\vec A=\dfrac{Q_{enclosed}}{\epsilon_0}$, relating the net electric flux through any closed surface to the net charge $Q_{enclosed}$ it encloses; true for both discrete point charges and continuous charge distributions, and it implies electric field lines start on positive charge and end on negative charge, so isolated positive or negative charges genuinely exist. Second equation -- Gauss's law for magnetism: $\displaystyle\oint\vec B\cdot d\vec A=0$, stating the net magnetic flux through any closed surface is always exactly zero, for every possible surface; this forces magnetic field lines to always close on themselves in continuous loops, which is the precise mathematical statement that no isolated magnetic monopole (a lone north or south magnetic charge) exists in nature -- if one did exist, the right-hand side would no longer be zero and this equation would need to be modified. Third equation -- Faraday's law of electromagnetic induction: $\displaystyle\oint\vec E\cdot d\vec l=-\dfrac{d\Phi_B}{dt}$, stating the line integral of the electric field around any closed path equals the negative rate of change of magnetic flux through the surface it bounds; this single law underlies every electric generator and transformer in use today. Fourth equation -- the modified Ampere's circuital law, also called the Ampere-Maxwell law: $\displaystyle\oint\vec B\cdot d\vec l=\mu_0 i_C+\mu_0\epsilon_0\dfrac{d\Phi_E}{dt}$, stating the line integral of the magnetic field around any closed path is produced jointly by the conduction current $i_C$ and Maxwell's displacement-current correction term through the surface bounded by that path. Combined, these four equations mathematically guarantee that a changing $E$-field and a changing $B$-field can continuously regenerate one another and propagate as a self-sustaining electromagnetic wave -- the complete theoretical foundation from which the existence, speed, and transverse nature of electromagnetic waves all follow.