Match the LIST-I with LIST-II Choose the correct answer from the options given below:LIST-I LIST-II A. $\nabla \cdot \bar{D} = \rho_v$ I. $\oint_S \bar{D} \cdot d\bar{s} = \int_V \rho_v dv$ B. $\nabla \cdot \bar{B} = 0$ II. $\oint_S \bar{B} \cdot d\bar{s} = 0$ C. $\nabla \times \bar{E} = -\frac{\partial \bar{B}}{\partial t}$ III. $\oint_L \bar{E} \cdot d\bar{l} = -\frac{\partial}{\partial t} \int_S \bar{B} \cdot d\bar{s}$ D. $\nabla \times \bar{H} = \bar{J} + \frac{\partial \bar{D}}{\partial t}$ IV. $\oint_L \bar{H} \cdot d\bar{l} = \int_S (J + \frac{\partial \bar{D}}{\partial t}) \cdot d\bar{s}$
Matching Maxwell's Equations Forms
This question requires matching the differential forms of Maxwell's equations (LIST-I) with their corresponding integral forms (LIST-II).
Understanding the Equations
- A. $\nabla \cdot \bar{D} = \rho_v$: This is Gauss's Law for Electricity (in terms of electric displacement field $\bar{D}$). It relates the divergence of $\bar{D}$ to the volume charge density $\rho_v$. Its integral form, derived using the Divergence Theorem, is I. $\oint_S \bar{D} \cdot d\bar{s} = \int_V \rho_v dv$, stating that the electric flux through a closed surface equals the enclosed charge. Thus, A matches with I.
- B. $\nabla \cdot \bar{B} = 0$: This is Gauss's Law for Magnetism. It states that the divergence of the magnetic field $\bar{B}$ is zero, implying the absence of magnetic monopoles. The integral form is II. $\oint_S \bar{B} \cdot d\bar{s} = 0$, meaning the net magnetic flux through any closed surface is zero. Thus, B matches with II.
- C. $\nabla \times \bar{E} = -\frac{\partial \bar{B}}{\partial t}$: This is Faraday's Law of Induction in differential form. It relates the curl of the electric field $\bar{E}$ to the time rate of change of the magnetic field $\bar{B}$. Its integral form is III. $\oint_L \bar{E} \cdot d\bar{l} = -\frac{\partial}{\partial t} \int_S \bar{B} \cdot d\bar{s}$, connecting the electromotive force (EMF) around a loop to the change in magnetic flux through the enclosed surface. Thus, C matches with III.
- D. $\nabla \times \bar{H} = \bar{J} + \frac{\partial \bar{D}}{\partial t}$: This is the Ampère-Maxwell Law. It relates the curl of the magnetic field intensity $\bar{H}$ to the conduction current density $\bar{J}$ and the time rate of change of the electric displacement field $\bar{D}$ (displacement current density). The integral form is IV. $\oint_L \bar{H} \cdot d\bar{l} = \int_S (\bar{J} + \frac{\partial \bar{D}}{\partial t}) \cdot d\bar{s}$, linking the line integral of $\bar{H}$ around a loop to the total current through the enclosed surface. Thus, D matches with IV.
Final Matches
Based on the analysis of each equation:
- A matches with I
- B matches with II
- C matches with III
- D matches with IV
This corresponds to the option A-I, B-II, C-III, D-IV.
Important Questions from Maxwell's Equations
∇ × H = J is differential form of
Maxwell's divergence equation for the magnetic field is given by _______.
If flux density is represented by 'B' and magnetic field is represented by 'H' in a magnetic circuit, then what will be the energy density in the magnetic field?
Maxwell's third equation is derived from _______.
Which law is represented by the given expression?
\(\int B.dl = \mu_oi_c+\mu_0\epsilon_0 \frac{d \Phi_E}{dt}\)
