question_answer

Directions : (6-10)

Gauss’s Low for Magnetism

By analogy to Guass's law of electrostatics, we can

write Gauss's law of magnetism as \[\oint{\overrightarrow{B}\,.\,\overrightarrow{ds}}={{\mu }_{0}}{{m}_{inside}}\] wehre \[\oint{\overrightarrow{B}\,.\,\overrightarrow{ds}}\] is the magnetic flux and \[{{m}_{inside}}\]is the net pole strength inside the closed surface.

We do not have an isolated magnetic pole in nature. At least none has been found to exist till date. The smallest unit of the source of magnetic field is a magnetic dipole where the net magnetic pole is zero. Hence, the net magnetic pole enclosed by any closed surface is always zero. Correspondingly, the flux of the magnetic field through any closed surface is zero.

Consider the two idealised systems

(i) a parallel plate capacitor with large plates and small separation and

(ii) a long solenoid of length L > > R, radius of cross-section

In (i) \[\overrightarrow{E}\] is ideally treated as a constant betwen plates and zero outside. In (ii) magnetic field is constant inside the solenoid and zero ouside. These idealised assumption, however, contradict fundamental laws as

A) case (i) contradicts Gauss's law for electrostatic field

B) case (ii) contradicts Gauss's law for magnetic fields

C) case (i) agrees with \[\oint{\overrightarrow{E}\,.\,\overrightarrow{dl}}=0\]

D) case (ii) contradicts \[\oint{\overrightarrow{H}\,.\,\overrightarrow{dl}}={{I}_{en}}\]