Question

Match List I with List II

Choose the correct answer from the options given below:

• A. A-I, B-III, C-IV, D-II
• B. A-III, B-IV, C-I, D-II
• C. A-IV, B-II, C-III, D-I
• D. A-II, B-III, C-IV, D-I

Answer 1

The Correct Option is D

The problem asks to match the laws of electromagnetism (List-I) with their corresponding equations (List-II). These laws are derived from Maxwell’s equations.

Gauss’s Law of Electrostatics states that the total electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space. The corresponding equation is:

\[ \oint \mathbf{E} \cdot d\mathbf{a} = \frac{1}{\epsilon_0} \int \rho dV. \]

This matches D – I.

Faraday’s Law of Electromagnetic Induction states that the electromotive force induced in a closed loop is equal to the negative rate of change of the magnetic flux through the loop. The corresponding equation is:

\[ \oint \mathbf{E} \cdot d\mathbf{l} = - \frac{d}{dt} \int \mathbf{B} \cdot d\mathbf{a}. \]

This matches B – III.

Ampere’s Law (with Maxwell’s correction) relates the line integral of the magnetic field around a closed loop to the current passing through the loop and the displacement current. For steady currents, the corresponding equation is:

\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I. \]

This matches C – IV.

Gauss’s Law for Magnetostatics states that the net magnetic flux through a closed surface is zero, indicating there are no magnetic monopoles. The corresponding equation is:

\[ \oint \mathbf{B} \cdot d\mathbf{a} = 0. \]

This matches A – II.

Final Matching:

A – II, B – III, C – IV, D – I

Answer 2

This problem requires matching fundamental laws of electromagnetism (List-I) with their corresponding mathematical integral forms (List-II). We will analyze each law and its mathematical representation to find the correct pairs.

Concept Used:

The four laws listed are part of Maxwell's equations, which form the foundation of classical electromagnetism.

• Gauss's Law of Electrostatics: This law states that the net electric flux through any closed surface is directly proportional to the total electric charge enclosed within that surface. The integral form is: \[ \oint_S \vec{E} \cdot d\vec{a} = \frac{Q_{\text{enc}}}{\epsilon_0} \] where \( Q_{\text{enc}} \) is the total enclosed charge. If the charge is distributed with a volume charge density \( \rho \), then \( Q_{\text{enc}} = \int_V \rho dV \). The law becomes: \[ \oint_S \vec{E} \cdot d\vec{a} = \frac{1}{\epsilon_0} \int_V \rho dV \]
• Gauss's Law of Magnetostatics: This law states that the net magnetic flux through any closed surface is always zero. This is a consequence of the fact that magnetic monopoles (isolated north or south poles) have never been observed to exist. Magnetic field lines always form closed loops. The integral form is: \[ \oint_S \vec{B} \cdot d\vec{a} = 0 \]
• Faraday's Law of Electromagnetic Induction: This law states that a changing magnetic flux through a surface induces an electromotive force (EMF) in any closed loop that bounds the surface. The EMF is equal to the negative of the rate of change of magnetic flux. The integral form is: \[ \oint_L \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} = -\frac{d}{dt} \int_S \vec{B} \cdot d\vec{a} \]
• Ampere's Circuital Law: This law states that the line integral of the magnetic field \( \vec{B} \) around any closed loop is proportional to the total electric current passing through the surface enclosed by the loop. The integral form is: \[ \oint_L \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \]

Step-by-Step Solution:

We will now match each law from List-I with its corresponding mathematical expression from List-II based on the concepts above.

Step 1: Match 'A. Gauss's law of magnetostatics'

As per the concept, Gauss's law for magnetism states that the net magnetic flux through a closed surface is zero. This corresponds to the mathematical expression:

\[ \oint \vec{B} \cdot d\vec{a} = 0 \]

This expression is listed as II in List-II. Therefore, A matches with II.

Step 2: Match 'B. Faraday's law of electromagnetic induction'

Faraday's law relates the induced EMF (line integral of \( \vec{E} \)) to the rate of change of magnetic flux (time derivative of the surface integral of \( \vec{B} \)). The equation is:

\[ \oint \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \int \vec{B} \cdot d\vec{a} \]

This expression is listed as III in List-II. Therefore, B matches with III.

Step 3: Match 'C. Ampere's law'

Ampere's law relates the line integral of the magnetic field around a closed loop to the enclosed current. The equation is:

\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I \]

This expression corresponds to IV in List-II. (Note: The negative sign in the question, \( \oint \vec{B} \cdot d\vec{l} = -\mu_0 I \), would arise from a specific choice of direction for the line element \(d\vec{l}\) opposite to the convention of the right-hand rule, but it represents the same physical law). Therefore, C matches with IV.

Step 4: Match 'D. Gauss's law of electrostatics'

Gauss's law for electrostatics relates the electric flux through a closed surface to the enclosed charge. The equation is:

\[ \oint \vec{E} \cdot d\vec{a} = \frac{1}{\epsilon_0} \int \rho dV \]

This expression is listed as I in List-II. Therefore, D matches with I.

Final Computation & Result:

The correct matching is as follows:

• A \(\rightarrow\) II
• B \(\rightarrow\) III
• C \(\rightarrow\) IV
• D \(\rightarrow\) I

This corresponds to the combination: A-II, B-III, C-IV, D-I.