Question

Match List I with List II

LIST I		LIST II
A	Gauss's Law in Electrostatics	I	\(\oint \vec{E} \cdot d \vec{l}=-\frac{d \phi_B}{d t}\)
B	Faraday's Law	II	\(\oint \vec{B} \cdot d \vec{A}=0\)
C	Gauss's Law in Magnetism	III	\(\oint \vec{B} \cdot d \vec{l}=\mu_0 i_c+\mu_0 \in_0 \frac{d \phi_E}{d t}\)
D	Ampere-Maxwell Law	IV	\(\oint \vec{E} \cdot d \vec{s}=\frac{q}{\epsilon_0}\)

Choose the correct answer from the options given below:

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

Answer 1

The Correct Option is B

The given question involves matching equations from electromagnetic theory to their respective physical laws. We will examine each law and its corresponding mathematical representation.

1. Gauss's Law in Electrostatics: This law relates the electric field to the charge enclosed by a surface. It states that the electric flux through a closed surface is proportional to the enclosed charge. Mathematically, it is expressed as: \(\oint \vec{E} \cdot d \vec{s} = \frac{q}{\epsilon_0}\). This matches with IV.
   Match: A-IV
2. Faraday's Law: Faraday's Law of electromagnetic induction states that the induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. It is given by the equation: \(\oint \vec{E} \cdot d \vec{l} = -\frac{d \phi_B}{d t}\). This matches with I.
   Match: B-I
3. Gauss's Law in Magnetism: This law states that the net magnetic flux out of any closed surface is zero (implying that magnetic monopoles do not exist). The equation is: \(\oint \vec{B} \cdot d \vec{A} = 0\). This matches with II.
   Match: C-II
4. Ampere-Maxwell Law: This law extends Ampere's Law to include the displacement current. It is mathematically expressed as: \(\oint \vec{B} \cdot d \vec{l} = \mu_0 i_c + \mu_0 \epsilon_0 \frac{d \phi_E}{d t}\). This matches with III.
   Match: D-III

Thus, the correct set of matches from the list options is:

Option: A-IV, B-I, C-II, D-III.