Magnetism and Matter Class 12 Physics Notes

• Torque on a dipole in a uniform field:
• $\tau =mB\sin \theta$
• (vector:
• $\vec{\tau }=\vec{m}\times \vec{B}$
• ).
• Potential energy:
• $U=-mB\cos \theta$
• (minimum at θ = 0, aligned).
• A compass needle aligns with the field because torque tends to align the moment with B.

• Earth behaves like a magnetic dipole with the magnetic south pole near the geographic north.
• Magnetic declination: angle between true north and magnetic north.
• Angle of dip: angle the total field makes with the horizontal.
• Horizontal component:
• $B_H=B\cos \delta$
• , vertical component:
• $B_V=B\sin \delta$
• .
• At the magnetic poles the dip is 90°; at the equator it is 0°.

• $\text{Diamagnetic:}$
• weakly repelled by a field; no unpaired electrons; susceptibility small and negative (e.g. bismuth, copper, water).
• $\text{Paramagnetic:}$
• weakly attracted; unpaired electrons; small positive susceptibility that decreases with temperature (e.g. aluminium, oxygen).
• $\text{Ferromagnetic:}$
• strongly attracted; domains align in the field; large positive susceptibility, exhibits hysteresis (e.g. iron, cobalt, nickel).

A ferromagnetic material's magnetisation lags behind the applied field, tracing a closed loop called the hysteresis curve. The area of the loop represents energy loss per cycle.

• Soft magnetic materials (e.g. iron): narrow hysteresis loop, low energy loss — used in transformer cores.
• Hard magnetic materials (e.g. steel): wide loop, retain magnetisation — used for permanent magnets.

Example: A bar magnet of moment $m=1{\text{A m}}^2$ is placed at 30° to a $B=0.2\text{T}$ field. Find the torque.

Solution: $\tau =mB\sin \theta =1\times 0.2\times \sin 30=0.2\times 0.5=0.1\text{N m}$.

Exercises (NCERT Q5.1–Q5.7)

• 5.1) A short bar magnet placed with its axis at 30° with a uniform external magnetic field of 0.25 T experiences a torque of magnitude equal to 4.5 × 10⁻² J. What is the magnitude of magnetic moment of the magnet?
• 5.2) A short bar magnet of magnetic moment m = 0.32 J T⁻¹ is placed in a uniform magnetic field of 0.15 T. If the bar is free to rotate in the plane of the field, which orientation would correspond to its (a) stable, and (b) unstable equilibrium? What is the potential energy of the magnet in each case?
• 5.3) A closely wound solenoid of 800 turns and area of cross section 2.5 × 10⁻⁴ m² carries a current of 3.0 A. Explain the sense in which the solenoid acts like a bar magnet. What is its associated magnetic moment?
• 5.4) If the solenoid in Exercise 5.5 is free to turn about the vertical direction and a uniform horizontal magnetic field of 0.25 T is applied, what is the magnitude of torque on the solenoid when its axis makes an angle of 30° with the direction of applied field? (Note: the printed cross-reference 'Exercise 5.5' points at the solenoid of Exercise 5.3.)
• 5.5) A bar magnet of magnetic moment 1.5 J T⁻¹ lies aligned with the direction of a uniform magnetic field of 0.22 T. (a) What is the amount of work required by an external torque to turn the magnet so as to align its magnetic moment: (i) normal to the field direction, (ii) opposite to the field direction? (b) What is the torque on the magnet in cases (i) and (ii)?
• 5.6) A closely wound solenoid of 2000 turns and area of cross-section 1.6 × 10⁻⁴ m², carrying a current of 4.0 A, is suspended through its centre allowing it to turn in a horizontal plane. (a) What is the magnetic moment associated with the solenoid? (b) What is the force and torque on the solenoid if a uniform horizontal magnetic field of 7.5 × 10⁻² T is set up at an angle of 30° with the axis of the solenoid?
• 5.7) A short bar magnet has a magnetic moment of 0.48 J T⁻¹. Give the direction and magnitude of the magnetic field produced by the magnet at a distance of 10 cm from the centre of the magnet on (a) the axis, (b) the equatorial lines (normal bisector) of the magnet.

Answers at a glance

1. 1
   Q5.1 → m = 0.36 J T⁻¹.
2. 2
   Q5.2 → Stable: m parallel to B, U = –0.048 J. Unstable: m anti-parallel to B, U = +0.048 J.
3. 3
   Q5.3 → The solenoid's field mimics a bar magnet with the grip-rule end as its north pole; m = NIA = 0.6 J T⁻¹.
4. 4
   Q5.4 → τ = 7.5 × 10⁻² N m.
5. 5
   Q5.5 → (a) Work: 0.33 J to reach 90°, 0.66 J to reach 180°. (b) Torque: 0.33 N m at 90°, zero at 180°.
6. 6
   Q5.6 → (a) m = 1.28 J T⁻¹. (b) Force = 0; torque = 4.8 × 10⁻² N m.
7. 7
   Q5.7 → (a) Axis: 9.6 × 10⁻⁵ T along m (S → N). (b) Equator: 4.8 × 10⁻⁵ T opposite to m (N → S).

MCQ Questions

1 Mark Questions

1. 1
   State Gauss's law in magnetism.
2. 2
   Define the magnetic dipole moment of a bar magnet and give its SI unit.
3. 3
   What are the angles of dip and declination at a place?
4. 4
   Give one example each of a diamagnetic and a paramagnetic material.

2 Mark Questions

1. 1
   A bar magnet of moment 0.5 A m² is placed in a uniform magnetic field of 0.2 T. Find the torque on it when it is inclined at 30° to the field. (Ans: τ = mB sin 30° = 0.05 N m)
2. 2
   The horizontal component of the earth's magnetic field at a place is 0.3 G and the angle of dip is 60°. Find the total field at that place. (Ans: B = 0.6 G)
3. 3
   Write the expression for the magnetic field at a point on the axial line of a short bar magnet and state its direction. (Ans: B = (μ₀/4π)·2m/r³, along the axis)

3 Mark Questions

1. 1
   Derive the expression for the magnetic field at a point on the equatorial line of a short bar magnet and compare it with the axial field. (Ans: B = (μ₀/4π)·m/r³, opposite to the direction of m; axial is double)
2. 2
   Explain the hysteresis loop for a ferromagnetic material. Distinguish between soft and hard magnetic materials with examples.
3. 3
   Distinguish between diamagnetic, paramagnetic and ferromagnetic substances by giving the signs and magnitudes of magnetic susceptibility and one example of each.

5 Mark Questions

1. 1
   Derive the magnetic field at a point on the axial line and on the equatorial line of a bar magnet, and show how a bar magnet behaves as a magnetic dipole with the torque τ = m × B.
2. 2
   Describe the earth's magnetism: explain magnetic declination, angle of dip and the horizontal component of the earth's field, and discuss the proposed sources of the earth's magnetism.