CBSE Class 12 Physics Chapter 5: Magnetism and Matter NCERT Solutions

A toroid is essentially a solenoid bent into a circular shape. The magnetic field lines inside a toroid are concentric circles and are confined within the toroid itself. For any point outside the toroid, the net current enclosed by a circular loop is zero. According to Ampere's law, the magnetic field outside the toroid is therefore zero. Since there is no net magnetic field outside, and the field inside is confined, the toroid does not behave like a magnetic dipole at large distances. A magnetic dipole would produce a field that falls off as $\backslash frac\{1\}\{r^3\}$ at large distances. As the magnetic field outside is zero, the net magnetic moment of the toroid is zero.

Answer: (c) is zero, otherwise there would be a field falling as $\backslash frac\{1\}\{r^3\}$ at large distances outside the toroid.

The Earth's magnetic field can be approximated by a magnetic dipole located at its center. The axis of this dipole is not perfectly aligned with the Earth's rotational axis; it is tilted by approximately 11.3°. Magnetic declination is the angle between the geographic north and the magnetic north at a given location. Due to the tilt of the dipole axis, this angle varies across the Earth's surface. At some locations, the magnetic north aligns closely with the geographic north (declination near zero), while at others, there is a significant angular difference. The maximum possible deviation from true north due to this tilt is the tilt angle itself. Therefore, the declination can range from 0° up to approximately 11.3° in both the West and East directions.

Answer: (a) the declination varies between 11.3° W to 11.3° E.

Permanent magnets are made from ferromagnetic materials. In these materials, atoms possess a net magnetic dipole moment due to the spin and orbital motion of electrons. These atomic magnetic moments interact with each other, causing them to align spontaneously over small regions called magnetic domains. In a permanent magnet, these domains are predominantly aligned in the same direction, resulting in a significant net magnetic moment for the material. While individual molecules (or atoms) have non-zero magnetic moments, it is the alignment of these moments within domains that defines a permanent magnet. Option (b) is incorrect because it implies individual molecules are perfectly aligned, which is not the primary characteristic; it's the domain alignment. Option (c) is incorrect because partial alignment leads to temporary magnetism or weaker magnetic properties, not a permanent magnet.

Answer: (d) domains are all perfectly aligned.

Let's analyze each option:

(a) Case (i): The assumption of a constant electric field E between large, closely spaced parallel plates and zero field outside is an idealization. While it simplifies calculations, it implies that the electric field lines terminate abruptly at the edges, which would contradict Gauss's law if applied to a surface enclosing the edges, as it would suggest a net charge density at the edges. However, the primary contradiction is not usually framed this way for Gauss's law for electrostatics itself, which holds true.

(b) Case (ii): The assumption of a constant magnetic field B inside a long solenoid and zero magnetic field outside is a common idealization. However, magnetic field lines are always closed loops; they do not begin or end. If the field is strictly zero outside, it implies that field lines must terminate at the solenoid's boundary, which contradicts Gauss's law for magnetic fields ($\backslash oint\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{A\}\; =\; 0$). This law states that the net magnetic flux through any closed surface is zero.

(c) Case (i): The integral $\backslash oint\; \backslash mathbf\{E\}\; \backslash cdot\; d\backslash mathbf\{l\}$ around a closed loop represents the work done by the electric field. For a static electric field (conservative field), this integral is zero. The idealized constant field between capacitor plates does not inherently contradict this; it's a property of static fields.

(d) Case (ii): The law $\backslash oint\; \backslash mathbf\{H\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; I\_\{en\}$ (or $\backslash oint\; \backslash mathbf\{B\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; \backslash mu\_0\; I\_\{en\}$) is Ampere's Law (or Ampere-Maxwell's Law). For an ideal solenoid, Ampere's law is used to derive the magnetic field inside. The contradiction arises not from Ampere's law itself but from the consequence of the idealized field distribution (zero outside) violating Gauss's law for magnetism.

Therefore, the assumption of a constant magnetic field inside and zero outside a solenoid contradicts Gauss's law for magnetic fields.

Answer: (b) case (ii) contradicts Gauss' law for magnetic fields.