Unit of Magnetic Flux — Weber (Wb)

Magnetic Flux (Φ) is defined as the total number of magnetic field lines passing through a surface perpendicularly.

Formula: Φ = B · A · cos θ

Where: • Φ (phi) = magnetic flux (SI unit: Weber, Wb) • B = magnetic flux density / magnetic field strength (SI unit: Tesla, T) • A = area of the surface (SI unit: m²) • θ = angle between the magnetic field (B) and the normal to the surface

Special cases: • θ = 0° (B perpendicular to surface): Φ = BA (maximum flux) • θ = 90° (B parallel to surface): Φ = 0 (no flux through surface) • θ = 180°: Φ = -BA (flux in opposite direction)

Vector form: Φ = B⃗ · A⃗ (dot product of B and area vector)

For non-uniform fields: Φ = ∫∫ B⃗ · dA⃗ (surface integral)

Magnetic flux is a scalar quantity.

SI Unit: Weber (Wb) Named after: Wilhelm Eduard Weber (1804–1891), German physicist

1 Weber defined as: The magnetic flux through a circuit when a current changing at 1 A/s induces an EMF of 1 Volt.

Equivalent expressions of Weber: 1 Wb = 1 V·s (Volt-second) 1 Wb = 1 T·m² (Tesla × metre squared) 1 Wb = 1 kg·m²·s⁻²·A⁻¹

Verification: Wb = T·m² 1 T = 1 kg·s⁻²·A⁻¹ 1 Wb = 1 T·m² = 1 kg·s⁻²·A⁻¹·m² = 1 kg·m²·s⁻²·A⁻¹ ✓

CGS unit of magnetic flux: Maxwell (Mx) 1 Wb = 10⁸ Maxwell

Other unit: 1 Wb = 10⁸ line (lines of force in old terminology)

Small values: 1 mWb = 10⁻³ Wb (milliweber) 1 μWb = 10⁻⁶ Wb (microweber)

Deriving from Φ = B·A·cosθ (cosθ is dimensionless):

[Φ] = [B] × [A]

Dimensions of B (magnetic field / flux density): From F = qvB: [B] = [F] / ([q][v]) = [MLT⁻²] / ([AT][LT⁻¹]) = [MLT⁻²] / [AL] = [MT⁻²A⁻¹]

Dimensions of A (area): [A] = [L²]

Therefore: [Φ] = [MT⁻²A⁻¹][L²] [Φ] = [ML²T⁻²A⁻¹]

Dimensional Formula of Magnetic Flux = [M¹L²T⁻²A⁻¹]

Verification using Φ = V·t (from Faraday's law): [V·t] = [ML²T⁻³A⁻¹][T] = [ML²T⁻²A⁻¹] ✓

Note: Magnetic flux [ML²T⁻²A⁻¹] vs Energy [ML²T⁻²] — they differ only by factor of [A⁻¹].

Faraday's Law of Electromagnetic Induction: The EMF induced in a circuit is proportional to the rate of change of magnetic flux through the circuit.

Formula: EMF (ε) = -dΦ/dt

Where: • ε = induced EMF (Volts) • dΦ/dt = rate of change of magnetic flux (Wb/s = V) • Negative sign → Lenz's Law

Verification: [dΦ/dt] = [ML²T⁻²A⁻¹] / [T] = [ML²T⁻³A⁻¹] = [Volt] ✓

For N turns of a coil: ε = -N × dΦ/dt

Lenz's Law: The induced EMF always opposes the change in flux that caused it (conservation of energy).

Flux linkage: λ = NΦ (total flux through all N turns) Unit of flux linkage: Weber-turn (Wb) [λ] = [ML²T⁻²A⁻¹] (same as flux)

Self-inductance: L = NΦ / I → [L] = [ML²T⁻²A⁻²] = Henry (H)

Example 1: Calculate magnetic flux A circular loop of radius 10 cm is placed in a uniform magnetic field B = 0.5 T. The plane of the loop is perpendicular to B (θ = 0°). A = πr² = π × (0.1)² = 0.0314 m² Φ = B·A·cosθ = 0.5 × 0.0314 × cos0° = 0.0157 Wb = 15.7 mWb

Example 2: At an angle Same loop tilted so B makes 60° with the normal: Φ = B·A·cos60° = 0.5 × 0.0314 × 0.5 = 7.85 mWb

Example 3: Induced EMF from Faraday's Law Flux through a 200-turn coil changes from 0.05 Wb to 0.01 Wb in 0.5 s: ε = -N × ΔΦ/Δt = -200 × (0.01 - 0.05)/0.5 = -200 × (-0.04/0.5) = -200 × (-0.08) = +16 V

Applications of magnetic flux: • Transformers (flux linkage between primary and secondary) • Electric generators (changing flux induces EMF) • Magnetic sensors and Hall effect sensors • MRI machines (large controlled magnetic flux) • Wireless charging (time-varying flux induces charging current)