The earth's magnetic field at a point on its surface is usually characterised by three quantities: (a) declination (b) inclination or dip and (c) horizontal component of the field. These are known as the elements of the earth's magnetic field. At a place, angle between geographic meridian and magnetic meridian is defined as magnetic declination, whereas angle made by the earth's magnetic field with the horizontal in magnetic meridian is known as magnetic dip.

(i) In a certain place, the horizontal component of magnetic field is \(\frac{1}{\sqrt{3}}\) times the vertical component. The angle of dip at this place is 

(ii) The angle between the true geographic north and the north shown by a compass needle is called as

(iii) The angles of dip at the poles and the equator respectively are

(iv) A compass needle which is allowed to move in a horizontal plane is taken to a geomagnetic pole. It

(v) Select the correct statement from the following

By analogy to Gauss's law of electrostatics, we can write Gauss's law of magnetism as \(\oint \vec{B} \cdot d \vec{s}=\mu_{0} m_{\text {inside }}\) where \(\oint \vec{B} \cdot d \vec{s}\) is the magnetic flux and \(m_{\text {inside }}\) 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.

(I) Consider the two idealised systems
(i) a parallel plate capacitor with large plates and small separation and
(ii) a long solenoid oflength L >> R, radius of cross-section.
In (i) \(\vec{E}\) is ideally treated as a constant between plates and zero outside. In (ii) magnetic field is constant inside the solenoid and zerq outside. These idealised assumptions, however, contradict fundamental laws as below

(ii) The net magnetic flux through any closed surface, kept in a magnetic field is

(iii) A closed surface S encloses a magnetic dipole of magnetic moment 2ml. The magnetic flux emerging from the surface is

(iv) Which of the following is not a consequence of Gauss's law?

(v) The surface integral of a magnetic field over a surface

When the atomic dipoles are aligned partially or fully, there is a net magnetic moment in the direction of the field in any small volume of the material. The actual magnetic field inside material placed in magnetic field is the sum of the applied magnetic field and the magnetic field due to magnetisation. This field is called magnetic intensity (H).
\(H=\frac{B}{\mu_{0}}-M\)
where M is the magnetisation of the material, llo is the permittivity of vacuum and B is the total magnetic field. The measure that tells us how a magnetic material responds to an external field is given by a dimensionless quantity is appropriately called the magnetic susceptibility: for a certain class of magnetic materials, intensity of magnetisation is directly proportional to the magnetic intensity.
(i) Magnetization of a sample is

(ii) Identify the wrongly matched quantity and unit pair.

(iii) A bar magnet has length- 3 cm, cross-sectional area 2 cm² and magnetic moment 3 A m². The intensity of magnetisation of bar magnet is

(iv) A solenoid has core of a material with relative permeability 500 and its windings carry a current of 1 A. The number of turns of the solenoid is 500 per metre. The magnetization of the material is nearly

(v) The relative permeability of iron is 6000. Its magnetic susceptibility is

The magnetic field lines of the earth resemble that of a hypothetical magnetic dipole located at the centre of the earth. The axis of the dipole is presently tilted by approximately 11.3^o with respect to the axis of rotation of the earth.

The pole near the geographic North pole of the earth is called the North magnetic pole and the pole near the geographic South pole is called South magnetic pole.
(i) The strength of the earth's magnetic field varies from place to place on the earth's surface, its value being of the order of

(ii) A bar magnet is placed North-South with its North-pole due North. The points of zero magnetic field will be in which direction from centre of magnet?

(iii) The value of angle of dip is zero at the magnetic equator because on it

(iv) The angle of dip at a certain place, where the horizontal and vertical components of the earth's magnetic field are equal, is

(v) At a place, angle of dip is 30°. If horizontal component of earth's magnetic field is H, then the total intensity of magnetic field will be

The field of a hollow wire with constant current is homageneous
Curves in the graph shown give, as functions of radius distance r, the magnitude B of the magnetic field inside and outside four long wires a, b, c and d, carrying currents that are uniformly distributed across the cross sections of the wires. Overlapping portions of the plots are indicated by double labels.

(i) Which wire has the greatest magnitude of the magnetic field on the surface?

(ii) The current density in a wire a is

(iii) Which wire has the greatest radius?

(iv) A direct current I flows along the length of an infinitely long straight thin walled pipe, then the magnetic field is

(v) In a coaxial, straight cable, the central conductor and the outer conductor carry equal currents in opposite direction. The magnetic field is zero

In small fields, ferromagnetic materials typically have much larger susceptibility, and therefore larger permeability, than paramagnetic materials. Ferromagnetism results because of spontaneous, self-aligning, cooperative interaction among relatively large number of iron atoms in regions called domains. As a result of molecular interactions the molecular magnetic moments in each domain are aligned parallel to one another. In other words, each domain is spontaneously magnetized to saturation even in the absence of any external magnetic field. The directions of magnetization in different domains are random, so that the resultant magnetization is zero and the specimen is unmagnetized.
(i) If above specimen is placed inside a solenoid; and slowly the magnetic intensity H (=ni) is increased from zero.
(a) What changes will occur in specimen on increasing H?
(b) Write the formula of magnetic flux density \((\overrightarrow{\mathbf{B}})\) for the specimen inside current carrying solenoid.
(c) Graphs of \(\mathbf{B}, \mu_{0} \mathbf{H} \text { and } \mu_{0} \mathbf{I}\) as a function of H are drawn. Identify which of the above physical quantities are contributing curves 1,2 and 3.

(ii) Why saturation of paramagnetic substances can be attained only at low temperatures?​​​​