Chapter 5 – Magnetism And Matter

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 m

• (a)is non-zero and points in the z-direction by symmetry.
• (b)points along the axis of the tortoid ( m = m φˆ ).
• (c)is zero, otherwise there would be a field falling as at large r3 distances outside the toroid.
• (d)is pointing radially outwards.

The magnetic field of 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° with the axis of Earth. At Mumbai, declination is nearly zero. Then,

• (a)the declination varies between 11.3° W to 11.3° E.
• (b)the least declination is 0°. Magnetism and Matter
• (c)the plane defined by dipole axis and Earth axis passes through Greenwich.
• (d)declination averaged over Earth must be always negative.

In a permanent magnet at room temperature

• (a)magnetic moment of each molecule is zero.
• (b)the individual molecules have non-zero magnetic moment which are all perfectly aligned.
• (c)domains are partially aligned.
• (d)domains are all perfectly aligned.

Consider the two idealized 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) 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)case (i) contradicts Gauss’s law for electrostatic fields.
• (b)case (ii) contradicts Gauss’s law for magnetic fields.
• (c)case (i) agrees with E.dl = 0 .
• (d)case (ii) contradicts H.dl = I en

A paramagnetic sample shows a net magnetisation of 8 Am–1 when placed in an external magnetic field of 0.6T at a temperature of 4K. 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

• (a)Am –1
• (b)Am –1
• (c)6 Am –1
• (d)2.4 Am –1 .

S is the surface of a lump of magnetic material.

• (a)Lines of B are necessarily continuous across S.
• (b)Some lines of B must be discontinuous across S.
• (c)Lines of H are necessarily continuous across S.
• (d)Lines of H cannot all be continuous across S.

The primary origin(s) of magnetism lies in

• (a)atomic currents.
• (b)Pauli exclusion principle.
• (c)polar nature of molecules.
• (d)intrinsic spin of electron.

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

• (a)The H field in the solenoid is (nearly) unchanged but the B field decreases drastically.
• (b)The H and B fields in the solenoid are nearly unchanged.
• (c)The magnetisation in the core reverses direction.
• (d)The magnetisation in the core diminishes by a factor of about 108.

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

• (a)electrostatic field lines can end on charges and conductors have free charges.
• (b)lines of B can also end but conductors cannot end them.
• (c)lines of B cannot end on any material and perfect shielding is not possible.
• (d)shells of high permeability materials can be used to divert lines of B from the interior region.

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

• (a)is always zero.
• (b)can be zero at specific points.
• (c)can be positive or negative.
• (d)is bounded.

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 = 106 A/m. Calculate the magnetisation current IM.

Explain quantitatively the order of magnitude difference between the diamagnetic susceptibility of N2 (~5 × 10–9) (at STP) and Cu (~10–5). Magnetism and Matter

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 it’s magnetic moment?

Verify the Gauss’s law for magnetic field of a point dipole of dipole N moment m at the origin for the surface which is a sphere of radius R. ? ? 60°

Three identical bar magnets are rivetted together at centre in the same plane as shown in Fig. 5.1. This system is placed at rest in a slowly varying magnetic field. It is found that the system of magnets 60° does not show any motion. The north-south poles of one magnet is ? ? shown in the Fig. 5.1. Determine the poles of the remaining two.

This question refers to a figure in the original PDF.

Suppose we want to verify the analogy between electrostatic and magnetostatic by an explicit experiment. Consider the motion of Fig. 5.1

This question refers to a figure in the original PDF.

• (i)electric dipole p in an electrostatic field E and
• (ii)magnetic dipole 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′ for each piece?

Use (i) the Ampere’s law for H and (ii) continuity of lines of B, to conclude that inside a bar magnet,

• (a)lines of H run from the N pole to S pole, while
• (b)lines of 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 m = m kˆ . 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 χ, the magnetic susceptibility? Consider an H-atom. Guess an expression for χ, upto a constant by constructing a quantity of dimensions of χ, out of parameters of the atom: e, m, v, R and µ 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 χ ~ 10 –5 for many solid materials.

Assume the dipole model for earth’s magnetic field B which is given µ0 2m cos θ by B V = vertical component of magnetic field = 4π r3 µ0 sinθ m BH = Horizontal component of magnetic field = 4π r3 θ = 90° – lattitude as measured from magnetic equator. Find loci of points for which

• (i)B is minimum;
• (ii)dip angle is zero; and
• (iii)dip angle is ± 45°.

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 coils made each from identical wires of length L. C1 is circular (radius R ) and C2 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.