Question

One of Maxwell’s equations can be written as $$\oint \overrightarrow{B}\cdot d\overrightarrow{A}=0$$.
Which of the following statements follows directly from this statement? Select two answers.
A. Magnetic field lines must form closed loops.
B. Moving charges create magnetic fields.
C. There exist no monopoles.
D. Magnetic field is directed away from the north pole of a bar magnet and toward the south magnet.
E. Magnetic field strength decreases with increasing distance from the source.

Answer 1

Hint: Recall Maxwell's equations. Check from which of Maxwell's equations the given equation can be derived and what is the significance of the given equation.

Complete step by step answer:
The statements B, D and E are not derived from Maxwell’s laws.
The equation $$\oint \overrightarrow{B}\cdot d\overrightarrow{A}=0$$ can be derived from Maxwell’s second equation.
The physical significance of the given equation is that the magnetic flux induction through any closed area $$\overrightarrow{A}$$ is zero.
Hence, it can be concluded that the magnetic field lines going in and coming out of any closed volume are equal which shows that the magnetic field lines should form a closed loop.
This in return also proves that there must be dipole and monopole does not exist.
Hence, the correct statements for the given equation are given by options A and C.

So, the correct answer is “Option A and C”.

Additional Information:
The Maxwell’s equations given by the physicist James Clerk Maxwell and hence are named after him. He used these equations to prove that the light is an electromagnetic phenomenon.
Maxwell's equations are useful because they can be used to show the propagation of electric and magnetic fields with a constant speed.
The Maxwell’s second law is as follows:
$$\mathrm{\nabla }\cdot B=0$$
Integrate the above equation over an arbitrary volume V, we get,
$$\underset{V}{\oint }\mathrm{\nabla }\cdot B=0$$
Use Gauss divergence theorem to convert the volume integral to surface integral.
$$\underset{A}{\oint }B\cdot dA=0$$
$$\Rightarrow \underset{A}{\oint }\overrightarrow{B}\cdot d\overrightarrow{A}=0$$

Note:
The equation $$\oint \overrightarrow{B}\cdot d\overrightarrow{A}=0$$ is derived from Maxwell’s second law. This equation is true for any arbitrary area. Hence, Maxwell's equation is proved over all surfaces.