Q.Write the modified (Ampère-Maxwell) form of Ampère's circuital law, including the displacement current term, and explain in words the physical meaning of each term in it.

The law. $\oint \vec{B}\cdot d\vec{l} = \mu_0\left(I_c + \epsilon_0\frac{d\Phi_E}{dt}\right)$

Meaning of each term. $\oint\vec{B}\cdot d\vec{l}$ is the line integral of the magnetic field around a chosen closed loop -- physically, a measure of how strongly the field 'circulates' around that loop. $I_c$ is the ordinary conduction current (actual moving charge) passing through any surface bounded by the loop. $\epsilon_0\,d\Phi_E/dt$ is Maxwell's displacement current, the rate of change of electric flux $\Phi_E$ through that same surface, scaled by $\epsilon_0$. $\mu_0$, the permeability of free space, is the same proportionality constant met in the original (conduction-current-only) form of the law.

Why it is written as a SUM. Both terms genuinely contribute to producing the circulating magnetic field -- a region can have conduction current only (an ordinary wire, away from a capacitor), displacement current only (the gap inside a charging capacitor), or, in general, some of each -- and the corrected law adds their contributions together so the SAME total answer for $\oint\vec{B}\cdot d\vec{l}$ is obtained regardless of which surface, bounded by the same loop, is used to evaluate it (Example 1 and 2 work through the capacitor case in detail).

Write the corrected law and explain, term by term, what each symbol represents and why the two current terms are added.

• Writing the law with only $I_c$ and forgetting the displacement-current term entirely -- that is the ORIGINAL, incomplete Ampère's law.
• Treating $I_d$ as though it must always be zero except inside a capacitor -- it is a general term, simply zero wherever the electric flux happens not to be changing.