Magnetic Effects of Current and Magnetism Notes – Complete Chapter

Biot-Savart's Law

Definition

Gives magnetic field due to a small current element Idl at a point
dB = (μ₀/4π) × Idl sinθ / r² (where θ = angle between dl and r)
SI unit of B = Tesla (T); CGS unit = Gauss (G); 1 T = 10⁴ G
Dimensional formula of B: [MT⁻²A⁻¹]
μ₀ = 4π×10⁻⁷ T·m/A (permeability of free space); dimensionless when divided by μ₀
Direction of dB: ⊥ to both dl and r (use right-hand palm rule: curl fingers from dl to r, thumb → B)

dB = (μ₀/4π) × I dl × r̂ / r² (vector form: dB = μ₀I(dl × r̂)/(4πr²))

μ₀/4π = 10⁻⁷ T·m/A

Comparison: Biot-Savart vs Coulomb's Law

Both follow inverse square law; both are long-range forces
Both fields obey superposition principle
E is along position vector r (scalar source); B is ⊥ to both dl and r (vector source)
B has angle dependence (sinθ); E does not
B is zero at points along the direction of current element
Gauss Law → Ampere's Circuital Law (analogy)

Magnetic Field — Important Configurations

Configuration	Magnetic Field B
Straight finite wire (angles α, β)	B = (μ₀I/4πr)(sinα + sinβ)
Infinite straight wire	B = μ₀I/(2πr)
Semi-infinite wire	B = μ₀I/(4πr)
Circular loop — centre	B = μ₀I/(2R) = μ₀nI/(2R) for n turns
Circular loop — on axis at x	B = μ₀IR²/[2(R²+x²)^3/2]
Circular arc (angle θ rad) — centre	B = μ₀Iθ/(4πR)
Solenoid (n turns/length) — inside	B = μ₀nI = μ₀NI/L
Solenoid — at end	B = μ₀nI/2
Toroid	B = μ₀NI/(2πR_avg) inside; B = 0 outside
Infinite hollow cylinder — inside	B = 0
Infinite hollow cylinder — surface	B = μ₀I/(2πR)
Infinite solid cylinder — inside	B = μ₀Ir/(2πR²) = (μ₀J/2)r

⚡ Key Results for Circular Loops

Wire of length L bent into circular loop of n turns: R = L/(2πn); B = μ₀nI/2R = μ₀πn²I/L
If R = constant and I constant: B ∝ n (more turns → stronger field)
If L = constant and I constant: B ∝ n² (doubling turns quadruples field)
3 concentric loops with planes ⊥ to each other: B_net = √3 × B_each

Ampere's Circuital Law

∮B·dl = μ₀I_enclosed

B on LHS is due to ALL currents; I_enc on RHS is only the current enclosed by Amperian loop

Rules for Applying Ampere's Law

Draw a symmetric Amperian loop passing through the point where B is to be found
Find current enclosed by the Amperian loop
Angle between B and dl must be 0°, 90°, or 180° at all points
Magnitude of B must be same at all points on the Amperian loop
Current in same direction as loop → +I; opposite direction → −I
Always valid; but not always useful for calculation (needs symmetry)

Magnetic Force on Moving Charge

F = q(v × B) = qvB sinθ (θ = angle between v and B)

Direction: Fleming's Left Hand Rule (FBI rule)

Properties of Magnetic Force

Always ⊥ to velocity → does no work → KE = constant → speed = constant
Momentum is variable (direction changes)
Power = F·v = 0 (since F ⊥ v)
If v is ∥ or anti-∥ to B → F = 0 (θ = 0° or 180°)
If charge is at rest → F = 0 (magnetic force on stationary charge = 0)
Magnetic field can only change direction of motion, NOT speed

Path of Charge in Magnetic Field	Condition	Result
Straight line (unchanged)	v ∥ B or v anti-∥ B	F = 0; a = 0; v = const
Circle	v ⊥ B	Circular motion; r = mv/(qB); T = 2πm/(qB)
Helix	v at angle θ to B (0 < θ < 90°)	v sinθ → circular; v cosθ → straight (along B)

Radius: r = mv/(qB) = p/(qB) = √(2mKE)/(qB) = √(2mqV)/(qB)

Time period: T = 2πm/(qB) (independent of speed and radius!)

Frequency: f = qB/(2πm) (cyclotron frequency)

Pitch of helix: P = v cosθ × T = v cosθ × 2πm/(qB)

⚡ Key Statements on Electromagnetic Force

Charge at rest, zero EM force → magnetic field MAY be zero (not necessarily)
Charge at rest, non-zero EM force → electric field MUST be non-zero
Charge moving without deviation in EM field → E = 0 AND B = 0, OR qE = qvB (velocity selector)
Velocity selector condition: v = E/B (only this speed passes undeviated)

Force on Current-Carrying Conductors

Force on straight wire: F = BIL sinθ (θ = angle between I direction and B)

F = I(L × B) (vector form)

Force on closed loop in uniform B: F_net = 0 (but torque may exist)

Force between two parallel wires: F/L = μ₀I₁I₂/(2πd)

Parallel currents → attract; Anti-parallel currents → repel

Force on Different Wire Shapes in Uniform Field

Straight wire of length L: F = BIL sinθ
Semicircular arc of radius R: F = 2BIR (same as diameter wire)
Any curved wire between A and B: F = BI × (distance AB) → same as straight wire AB
Closed loop: F_net = 0

Torque on Current Loop and Moving Coil Galvanometer

Torque: τ = MB sinθ = NIAB sinθ (θ = angle between M and B)

τ = NI(A × B); M = NIA (magnetic moment; vector along axis of loop)

PE: U = −M·B = −MB cosθ

Net force on loop in uniform field = 0; net torque ≠ 0 (unless M ∥ B)

Moving Coil Galvanometer (MCG)

Working Principle: Torque on magnetic dipole in uniform magnetic field

At equilibrium: Deflecting torque = Restoring torque (spring)

NIAB = Cφ → φ = NIAB/C (φ = deflection angle; C = torsional constant of spring)

Current sensitivity (I_s): φ/I = NAB/C (larger → more sensitive)

Voltage sensitivity (V_s): φ/V = NAB/(CR) (R = coil resistance)

To increase sensitivity: increase N, A, B; decrease C (weaker spring)

Q: A rectangular loop (20 cm × 10 cm), 100 turns, carries 2 A in B = 0.5 T. Find max torque.

τ_max = NIAB = 100 × 2 × (0.20 × 0.10) × 0.5 = 100 × 2 × 0.02 × 0.5 = 2 N·m

Ans: τ_max = 2 N·m ✓

Bar Magnet — Magnetism and Matter

Bar Magnet Properties

Magnetic dipole: pole strength +m (N) and −m (S) separated by 2l
Magnetic moment M = m × 2l; vector from S to N; unit = A·m²
Pole strength m: scalar; unit = A·m
B field away from N pole, towards S pole
Same poles repel, opposite poles attract

Position	Magnetic Field B	Direction
Axial point (distance r, r >> l)	B = 2kM/r³ = μ₀M/(2πr³)	Along M (from S to N)
Equatorial point (distance r, r >> l)	B = kM/r³ = μ₀M/(4πr³)	Anti-parallel to M (from N to S)
General point at angle θ	B = (μ₀/4π) × M√(3cos²θ+1)/r³	At angle α: tanα = tanθ/2

Cutting a Bar Magnet

Cut along length (n equal parts along length): each part has same pole strength m, half length → M' = M/n (magnetic moment reduced)
Cut ⊥ to length (n equal parts): each part has same pole strength m, same pole area → M' = M/n; m' = m (pole strength same)
Bending bar magnet into semicircle: M' = M sin(θ/2) where θ = arc angle; for semicircle θ = 180°: M' = 2m (distance between poles = 2R)

Oscillation of Bar Magnet in Magnetic Field

Time period: T = 2π√(I/MB) (I = moment of inertia; M = magnetic moment; B = field)
Vibrational magnetometer: T = 2π√(I/MB_H) (B_H = horizontal component of Earth's field)
Tangent law: B_vertical/B_horizontal = tan θ
Stable equilibrium: θ = 0° (M ∥ B); Unstable: θ = 180°

Magnetic Properties of Materials

Key Magnetic Quantities

Magnetic Field Intensity H: H = B/μ − M; unit = A/m; H due to free current only
Magnetization I: magnetic moment per unit volume; unit = A/m
Magnetic Permeability μ: μ = B/H; scalar; unit = Wb/(A·m) = T·m/A
Relative Permeability μ_r: μ_r = μ/μ₀; dimensionless, no unit
Magnetic Susceptibility χ_m: χ_m = I/H; dimensionless; χ_m = μ_r − 1

Property	Diamagnetic	Paramagnetic	Ferromagnetic
Behaviour in external field	Weakly repelled; moves to weak field region	Weakly attracted; moves to strong field	Strongly attracted; moves to strong field
χ_m	Small negative (−1 ≤ χ_m < 0)	Small positive (0 < χ_m < ε)	Large positive (χ_m >> 1)
μ_r	Slightly < 1	Slightly > 1	Very >> 1
B_net	B_net < B₀	B_net > B₀ (slightly)	B_net >> B₀
Temperature effect	Independent of T	χ ∝ 1/T (Curie's Law)	Becomes paramagnetic above Curie temperature T_C
Mechanism	Universal; Lenz's law; paired electrons	Unpaired electrons; random orientation	Domain formation; unpaired electrons
Examples	Bismuth, Cu, Ag, Au, water, air, H₂, N₂	Al, Na, Mn, Pt, O₂, CuCl₂	Fe, Co, Ni, Gd

Curie's Law and Curie-Weiss Law

Curie's Law (paramagnetic): χ_m = C/T (susceptibility inversely proportional to temperature)
Curie-Weiss Law (ferromagnetic above T_C): χ_m = C/(T − T_C)
At T = T_C (Curie temperature): ferromagnetic → paramagnetic transition
Domains break apart above T_C → random alignment → paramagnetic behaviour

Hysteresis (I-H Curve for Ferromagnets)

Retentivity (OA): magnetic moment retained when H = 0 after magnetisation
Coercivity (OB): reverse field needed to demagnetize the material
Area of hysteresis loop = energy loss per cycle
Permanent magnet: high retentivity, high coercivity (e.g. steel, alnico)
Electromagnet core: low retentivity, high permeability (e.g. soft iron)

Q: A proton and α-particle enter perpendicular to a magnetic field B. If they have same KE, find ratio of their radii.

r = √(2mKE)/(qB); r_p/r_α = √(m_p/m_α) × (q_α/q_p) = √(1/4) × (2/1) = (1/2) × 2 = 1; same radius!

Ans: r_p = r_α (equal radii for same KE) ✓

Q: Two magnets oscillate in B_H. Alone: T₁ and T₂. Find ratio M₁/M₂.

T = 2π√(I/MB_H); M ∝ 1/T² (for same I); M₁/M₂ = T₂²/T₁²

Ans: M₁/M₂ = T₂²/T₁² ✓

Magnetic Effects & Magnetism