A toroid of $n$ turns, mean radius $R$ and cross-sectional radius $a$ carries current $I$. It is placed on a horizontal table taken as $x y$-plane. Its magnetic moment $\mathbf{m}$

The magnetic field of the earth can be modelled by that of a point dipole placed at the centre of the earth. The dipole axis makes an angle of $11.3^{\circ}$ with the axis of the earth. At Mumbai, declination is nearly zero. Then,

In a permanent magnet at room temperature,

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

A paramagnetic sample shows a net magnetisation of $8 \mathrm{Am}^{-1}$ when placed in an external magnetic field of 0.6 T at a temperature of 4 K . When the same sample is placed in an external magnetic field of 0.2 T at a temperature of 16 K , the magnetisation will be

$S$ is the surface of a lump of magnetic material.

The primary origin (s) of magnetism lies in

A long solenoid has 1000 turns per metre and carries a current of 1 A . It has a soft iron core of $\alpha_r=1000$. The core is heated beyond the Curie temperature, $T_c$.

Essential difference between electrostatic shielding by a conducting shell and magnetostatic shielding is due to

Let the magnetic field on the earth be modelled by that of a point magnetic dipole at the centre of the earth. The angle of dip at a point on the geographical equator

A proton has spin and magnetic moment just like an electron. Why then its effect is neglected in magnetism of materials?

A permanent magnet in the shape of a thin cylinder of length 10 cm has $M=10^6 \mathrm{~A} / \mathrm{m}$. Calculate the magnetisation current $I_M$.

Explain quantitatively the order of magnitude difference between the diamagnetic susceptibility of $\mathrm{N}_2\left(\sim 5 \times 10^{-9}\right)$ (at STP) and $\mathrm{Cu}\left(\sim 10^{-5}\right)$.

From molecular view point, discuss the temperature dependence of susceptibility for diamagnetism, paramagnetism and ferromagnetism.

A ball of superconducting material is dipped in liquid nitrogen and placed near a bar magnet.

(i) In which direction will it move?

(ii) What will be the direction of its magnetic moment?

Verify the Gauss's law for magnetic field of a point dipole of dipole moment m at the origin for the surface which is a sphere of radius $R$.

Three identical bar magnets are rivetted together at centre in the same plane as shown in figure. This system is placed at rest in a slowly varying magnetic field. It is found that the system of magnets does not show any motion. The north-south poles of one magnet is shown in the figure. Determine the poles of the remaining two.

Suppose we want to verify the analogy between electrostatic and magnetostatic by an explicit experiment. Consider the motion of (i) electric dipole $\mathbf{p}$ in an electrostatic field $\mathbf{E}$ and (ii) magnetic dipole $\mathbf{M}$ in a magnetic field B. Write down a set of conditions on E, B, p, M so that the two motions are verified to be identical. (Assume identical initial conditions).

A bar magnet of magnetic moment $M$ and moment of inertia $I$ (about centre, perpendicular to length) is cut into two equal pieces, perpendicular to length. Let $T$ be the period of oscillations of the original magnet about an axis through the mid-point, perpendicular to length, in a magnetic field $B$. What would be the similar period $T^{\prime}$ for each piece?

Use (i) the Ampere's law for H and (ii) continuity of lines of $\mathbf{B}$, to conclude that inside a bar magnet, (a) lines of $\mathbf{H}$ run from the N -pole to $S$ - pole, while (b) lines of $\mathbf{B}$ must run from the $S$-pole to $N$-pole.

Verify the Ampere's law for magnetic field of a point dipole of dipole moment $\mathbf{M}=M \hat{\mathbf{k}}$. Take $C$ as the closed curve running clockwise along

(i) the $z$-axis from $z=a>0$ to $z=R$,

(ii) along the quarter circle of radius $R$ and centre at the origin in the first quadrant of $x z$-plane,

(iii) along the $x$-axis from $x=R$ to $x=a$, and

(iv) along the quarter circle of radius $a$ and centre at the origin in the first quadrant of $x z$-plane

What are the dimensions of $\chi$, the magnetic susceptibility? Consider an H -atom. Gives an expression for $\chi$, upto a constant by constructing a quantity of dimensions of $\chi$, out of parameters of the atom $e, m, v, R$ and $\alpha_0$. Here, $m$ is the electronic mass, $v$ is electronic velocity, $R$ is Bohr radius. Estimate the number so obtained and compare with the value of $|\chi| \sim 10^{-5}$ for many solid materials.

Assume the dipole model for the earth's magnetic field $B$ which is given by $B_V=$ vertical component of magnetic field $=\frac{\alpha_0}{4 \pi} \frac{2 m \cos \theta}{r^3}$ $B_H=$ horizontal component of magnetic field $=\frac{\alpha_0}{4 \pi} \frac{\sin \theta m}{r^3}$ $\theta=90 \curlyvee$ - lattitude as measured from magnetic equator. Find loci of points for which (a) $|B|$ is minimum (b) dip angle is zero and (c) dip angle is $45^{\circ}$.

Consider the plane $S$ formed by the dipole axis and the axis of earth. Let $P$ be point on the magnetic equator and in $S$. Let $Q$ be the point of intersection of the geographical and magnetic equators. Obtain the declination and dip angles at $P$ and $Q$.

There are two current carrying planar coil made each from identical wires of length L. $C_1$ is circular (radius $R$ ) and $C_2$ is square (side $a$ ). They are so constructed that they have same frequency of oscillation when they are placed in the same uniform B and carry the same current. Find $a$ in terms of $R$.