Class 12 Physics Assignment

Magnetic dipole moment is defined as the product of pole strength and magnetic length of the magnet: M = m × 2l. Its SI unit is A m². Its direction is from south pole to north pole inside the magnet.

M = pole strength × distance between poles = 20 × 0.10 = 2 A m².

For a short magnet of magnetic moment M, at an axial point distance r from centre, B = μ₀/(4π) × 2M/r³. This field is along the magnetic moment direction.

For a short magnet, the magnitude of magnetic field at an equatorial point distance r from centre is B = μ₀/(4π) × M/r³. Vector form includes a negative sign because the direction is opposite to the magnetic moment.

B = 10⁻⁷ × 2M/r³ = 10⁻⁷ × 1.6/(0.2)³ = 1.6×10⁻⁷/0.008 = 2×10⁻⁵ T.

B = 10⁻⁷ × M/r³ = 10⁻⁷ × 0.8/0.008 = 1×10⁻⁵ T.

In magnitude, B_axial = 2B_equatorial at the same distance. Axial field is along magnetic moment; equatorial field is opposite to magnetic moment.

Torque τ = MB sinθ. Potential energy U = −MB cosθ.

τ = MB sinθ = 5 × 0.2 × sin30° = 1 × 0.5 = 0.5 N m.

Work done = ΔU = MB(cosθ₁ − cosθ₂) = 3×0.4(cos0° − cos90°) = 1.2 J.

Potential energy U = −MB cosθ is minimum when cosθ = 1, so θ = 0°.

Potential energy is maximum when θ = 180°, therefore equilibrium is unstable.

At an axial point on the north side, magnetic field is away from the north pole, i.e. towards right.

Magnetic field lines always form closed loops and every magnet has both north and south poles. Cutting a magnet produces smaller dipoles, not isolated poles. Hence magnetic monopoles have not been observed.

Gauss's law in magnetism: ∮B·dA = 0. It means net magnetic flux through any closed surface is zero because magnetic monopoles do not exist.

Net magnetic flux through a closed surface is zero. Therefore flux entering = flux leaving.

Magnetic declination is the angle between geographic meridian and magnetic meridian at a given place.

Angle of dip is the angle made by Earth's resultant magnetic field with the horizontal direction at a place.

B_H = B cosδ. So B = B_H/cos60° = 3×10⁻⁵/0.5 = 6×10⁻⁵ T.

B_H = B cosδ = 5×10⁻⁵×0.8 = 4×10⁻⁵ T. B_V = B sinδ = 5×10⁻⁵×0.6 = 3×10⁻⁵ T.

B_H = B cosδ, B_V = B sinδ, tanδ = B_V/B_H, and B = √(B_H² + B_V²).

At magnetic equator, Earth's field is horizontal, hence angle of dip is zero.

At magnetic poles, Earth's magnetic field is vertical, so dip angle is 90°.

Magnetization I is magnetic dipole moment per unit volume: I = M/V. SI unit is A m⁻¹.

I = magnetic moment/volume = 0.02/(5×10⁻⁶) = 4×10³ A/m.

Magnetic susceptibility χ measures how easily material gets magnetized: I = χH. Relative permeability μᵣ = μ/μ₀. Relation: μᵣ = 1 + χ.

μᵣ = 1 + χ = 1 + 0.002 = 1.002.

μᵣ = 1 + χ, so χ = μᵣ − 1 = 800 − 1 = 799.

Diamagnetic: χ is small negative. Paramagnetic: χ is small positive. Ferromagnetic: χ is large positive and depends on temperature and magnetization history.

Diamagnetic: bismuth, copper. Paramagnetic: aluminium, oxygen. Ferromagnetic: iron, cobalt, nickel.

In diamagnetic substances, induced magnetic moment develops opposite to applied magnetic field. Hence they are weakly repelled by strong magnetic field.

Paramagnetic substances have permanent atomic magnetic dipoles. In external field, these dipoles partially align with the field, causing weak attraction.

Curie's law: χ = C/T, where χ is magnetic susceptibility, C is Curie constant, and T is absolute temperature. Susceptibility decreases as temperature increases.

By Curie's law, χ ∝ 1/T. χ₂ = χ₁T₁/T₂ = 4×10⁻⁵×300/600 = 2×10⁻⁵.

Curie temperature is the temperature above which a ferromagnetic substance loses ferromagnetic behaviour and becomes paramagnetic.

Hysteresis is the lagging of magnetization or magnetic induction behind the magnetizing field during cyclic magnetization. It is represented by B-H or I-H hysteresis loop.

Retentivity is residual magnetism left when magnetizing field is reduced to zero. Coercivity is the reverse magnetizing field required to reduce magnetization to zero.

Retentivity is the B-axis intercept when H = 0. Coercivity is the negative H-axis intercept where B becomes zero.

Permanent magnets require high retentivity and high coercivity so they retain magnetism and are difficult to demagnetize.

Transformer cores require soft ferromagnetic material with narrow hysteresis loop, high permeability and low hysteresis loss.

Soft iron has high permeability and low coercivity. It is easily magnetized and demagnetized, making it suitable for electromagnets.

Steel has high coercivity and good retentivity. Once magnetized, it retains magnetism for long time and is difficult to demagnetize.

For a solenoid, H = nI = 1000 × 2 = 2000 A/m.

B = μ₀μᵣnI = 4π×10⁻⁷ ×1000×800×5 = 1.256×10⁻³×4000 ≈ 5.03 T. This is a theoretical value for the given high relative permeability.

I = χH, so χ = I/H = 500/1000 = 0.5.

Potential energy U = −MB cosθ is minimum when θ = 0°, so magnetic moment points along B. Since M is from S to N, the north end should point along field direction.

τ = MB sin90° = 4×0.25 = 1 N m. U = −MB cos90° = 0.

Work done by external agent = ΔU = U₂ − U₁ = MB(cosθ₁ − cosθ₂). Thus W = 2×0.5(cos30° − cos150°) = 1(√3/2 + √3/2) = √3 J.

Magnetic field lines are continuous closed curves. Outside the magnet they emerge from north pole and enter south pole. Inside the magnet they complete the loop from south to north.

Axial field B = 10⁻⁷ × 2M/r³. M = Br³/(2×10⁻⁷) = 8×10⁻⁵ × (0.1)³ /(2×10⁻⁷) = 8×10⁻⁸/(2×10⁻⁷) = 0.4 A m².