Match List – I with List – II and select the correct answer using codes given below : Codes :List – I List – II a. \(\nabla\times\overline{H}\) i. \(\rho_{v}\) b. \(\nabla\cdot\overline{D}\) ii. \(-\dfrac{\partial B}{\partial t}\) c. \(\nabla\times\overline{E}\) iii. \(\overline{J}+\dfrac{\partial\overline{D}}{\partial t}\) d. \(\nabla\cdot\left(\nabla\times\overline{B}\right)\) iv. 0
a-iii, b-i, c-ii, d-iv
Three of these are Maxwell's equations and the fourth is a vector identity.
(a) \(\nabla\times\overline{H}=\overline{J}+\dfrac{\partial\overline{D}}{\partial t}\) → (iii). Ampere's law with Maxwell's displacement-current term: a magnetic field circulates around a conduction current and around a changing electric flux.
(b) \(\nabla\cdot\overline{D}=\rho_{v}\) → (i). Gauss's law for electricity: electric flux diverges from charge, so charge is a source of the D field.
(c) \(\nabla\times\overline{E}=-\dfrac{\partial\overline{B}}{\partial t}\) → (ii). Faraday's law of induction, the minus sign being Lenz's law — the induced effect opposes the change that caused it.
(d) \(\nabla\cdot\left(\nabla\times\overline{B}\right)=0\) → (iv). This one is not a Maxwell equation but a vector identity: the divergence of any curl is identically zero, whatever the field. It holds for every differentiable vector field, physical or not.
The order (iii), (i), (ii), (iv) is option 2.
| Expression | Equals | Name |
|---|---|---|
| \(\nabla\times\overline{H}\) | \(\overline{J}+\dot{\overline{D}}\) | Ampere-Maxwell |
| \(\nabla\cdot\overline{D}\) | \(\rho_{v}\) | Gauss, electric |
| \(\nabla\times\overline{E}\) | \(-\dot{\overline{B}}\) | Faraday |
| \(\nabla\cdot(\nabla\times\overline{B})\) | 0 | Vector identity |
Why entry (d) is worth its place. The identity is exactly what makes the magnetic vector potential possible. Since \(\nabla\cdot\overline{B}=0\) is one of Maxwell's equations, and since the divergence of a curl always vanishes, B can be written as
\(\overline{B}=\nabla\times\overline{A}\)
with no loss of generality — and that substitution is the starting point of antenna theory and of the retarded-potential solution for radiation.
Its companion identity is equally useful: \(\nabla\times\left(\nabla V\right)=0\), the curl of any gradient is zero. Because the electrostatic field has zero curl, it can be written as \(\overline{E}=-\nabla V\) — which is why a voltage exists at all in static problems, and why it does not in the presence of a changing magnetic field.
Hence, the correct match is a-iii, b-i, c-ii, d-iv.
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