🔢 Formula & Dimensional Analysis

Complete Formula Bank

Every formula in Electromagnetic Induction — with dimensions, units, variables, and exam shortcuts. Searchable. Copyable.

⚡ Core Formulas

Magnetic Flux

Magnetic Flux (General)

Φ = BA cosθ

[M L² T⁻² A⁻¹] = Wb = V·s = T·m²

B = magnetic field (T) | A = area (m²) | θ = angle between B and normal to area

💡 θ is with normal, NOT surface. Maximum when θ=0°, zero when θ=90°

Faraday's Law

Induced EMF (Faraday-Lenz)

ε = −N dΦ/dt

Unit: Volt (V) = Wb/s

N = number of turns | dΦ/dt = rate of flux change | Negative sign encodes Lenz's Law

💡 For average EMF: ε_avg = −NΔΦ/Δt. Magnitude only in numericals unless direction asked.

Motional EMF

Rod Moving in Magnetic Field

ε = BLv

Unit: Volt (V)

B = magnetic field (T) | L = length of rod (m) | v = velocity (m/s) | (for v ⊥ B)

💡 General: ε = BLv sinα where α = angle between v and B. If v not ⊥ B, component matters.

Motional EMF

Current in Rail Problem

I = BLv/R

Unit: Ampere (A)

R = total resistance of circuit | B = field | L = separation between rails | v = velocity of rod

💡 Multiple resistors in the circuit → use equivalent resistance. Rod itself may have resistance.

Motional EMF

Opposing Force on Rod

F = B²L²v/R

Unit: Newton (N)

This force acts opposite to velocity (Lenz's Law). Derivation: F = BIL = B(BLv/R)L

💡 Terminal velocity when applied force = opposing force: v_t = FR/B²L²

Motional EMF

Rotating Rod EMF

ε = ½BωL²

Unit: Volt (V) | ω in rad/s

ω = angular velocity | L = length of rod | B = uniform field ⊥ plane of rotation

💡 Derived by integrating dε = B(rω)dr from 0 to L. Appears in JEE Advanced.

Self-Inductance

Definition of Self-Inductance

ε = −L dI/dt

[M L² T⁻² A⁻²] = Henry (H) = Ω·s

L = self-inductance | dI/dt = rate of current change | ε = back-EMF induced

💡 Also: L = NΦ/I from flux linkage definition. Use this to calculate L for given geometry.

Self-Inductance

Solenoid Self-Inductance

L = μ₀N²A/l

Unit: Henry (H) | μ₀ = 4π × 10⁻⁷ H/m

N = total turns | A = cross-section area (m²) | l = length (m) | Also: L = μ₀n²Al where n = N/l

💡 For solenoid with core of relative permeability μᵣ: L = μ₀μᵣN²A/l

Mutual Inductance

Induced EMF (Secondary)

ε₂ = −M dI₁/dt

M in Henry (H) | Same dimensional formula as L

M = mutual inductance | dI₁/dt = rate of current change in primary | ε₂ = EMF in secondary

💡 M₁₂ = M₂₁ = M (symmetric). Also: N₂Φ₂₁ = MI₁ and N₁Φ₁₂ = MI₂

Mutual Inductance

Coaxial Solenoids

M = μ₀N₁N₂A/l

Unit: Henry (H)

N₁, N₂ = turns in each solenoid | A = area of smaller solenoid | l = length of solenoid

💡 Coupling coefficient: k = M/√(L₁L₂). Perfect coupling k=1. Practical transformers: k close to 1.

Energy

Energy Stored in Inductor

U = ½LI²

Unit: Joule (J) | Analogous to ½mv²

L = inductance (H) | I = steady current (A)

💡 Energy is stored in the magnetic field of the inductor, not in the wire. Released when circuit is broken.

Energy Density

Magnetic Energy Density

u_B = B²/(2μ₀)

Unit: J/m³

B = magnetic field (T) | μ₀ = 4π × 10⁻⁷ H/m | Per unit volume energy

💡 Compare with electric field energy density: u_E = ε₀E²/2. Both appear in EM waves chapter.

AC Generator

EMF of Rotating Coil

ε = NBAω sin(ωt)

ε₀ = NBAω = peak EMF

N = turns | B = field | A = area | ω = angular velocity | ε₀ = peak/maximum EMF

💡 Derived from: Φ = NBA cos(ωt) → ε = −dΦ/dt = NBAω sin(ωt). CBSE derivation must-do.

Transformer

Transformer Equation

V₂/V₁ = N₂/N₁ = I₁/I₂

Dimensionless ratios | Ideal transformer: 100% efficiency

V₁,V₂ = primary/secondary voltages | N₁,N₂ = turns | I₁,I₂ = currents | V₁I₁ = V₂I₂

💡 Step-up: N₂ > N₁ → V₂ > V₁, I₂ < I₁. Step-down: opposite. Power is conserved.

Power

Power Dissipated (Rail Problem)

P = B²L²v²/R

Unit: Watt (W) | P = εI = ε²/R

Derived from P = Fv = (B²L²v/R)·v = B²L²v²/R | Also = ε²/R = (BLv)²/R

💡 Work done by external agent = heat generated in circuit. Perfect energy conversion example.

Charge

Induced Charge (Ballistic)

q = NΔΦ/R

Unit: Coulomb (C)

Derived from: q = ∫I dt = ∫(ε/R)dt = (1/R)∫(NdΦ/dt)dt = NΔΦ/R

💡 Induced charge is INDEPENDENT of time taken to change flux. Used in ballistic galvanometer.

LC Circuits

Angular Frequency of LC Oscillation

ω = 1/√(LC)

Unit: rad/s | Time period T = 2π√(LC)

L = inductance | C = capacitance | Energy oscillates between ½LI²_max and ½CV²_max

💡 Maximum current: I_max = V_max √(C/L). Maximum charge: Q_max at I = 0.

Combined Inductance

Inductors in Series/Parallel

L_s = L₁+L₂ | 1/L_p = 1/L₁+1/L₂

Unit: Henry (H) | Similar to resistors but for inductance

Series: no mutual coupling assumed. Parallel: same. With mutual coupling: L_s = L₁+L₂±2M

💡 With mutual inductance: aiding (+2M) or opposing (−2M). JEE Advanced favorite.

📐 Dimensional Analysis Table

Quantity	Symbol	SI Unit	Dimensional Formula	Equivalent Units
Magnetic Flux	Φ	Weber (Wb)	[M L² T⁻² A⁻¹]	V·s = T·m²
Magnetic Field	B	Tesla (T)	[M T⁻² A⁻¹]	Wb/m² = kg/(A·s²)
Self-Inductance	L	Henry (H)	[M L² T⁻² A⁻²]	Wb/A = Ω·s = V·s/A
Mutual Inductance	M	Henry (H)	[M L² T⁻² A⁻²]	Same as L
EMF	ε	Volt (V)	[M L² T⁻³ A⁻¹]	J/C = W/A
Permeability	μ₀	H/m	[M L T⁻² A⁻²]	N/A² = T·m/A = Wb/(A·m)
Inductance/turn	L/N	H/turn	[M L² T⁻² A⁻²]	Wb/A = T·m²/A
Energy	U	Joule (J)	[M L² T⁻²]	V·A·s = W·s
Energy Density	u_B	J/m³	[M L⁻¹ T⁻²]	Pa = N/m²

🧮 Interactive Calculators

🌐 Magnetic Flux Calculator

B (Tesla):

A (m²):

θ (degrees):

⚡ Induced EMF Calculator (Faraday)

N (turns):

ΔΦ (Wb):

Δt (seconds):

🏃 Motional EMF Calculator (ε = BLv)

B (Tesla):

L (meters):

v (m/s):

🌀 Solenoid Inductance Calculator

N (total turns):

μᵣ (relative perm):

A (m²):

l (length, m):

⚡ Energy Stored in Inductor (U = ½LI²)

L (Henry):

I (Amperes):

🔬 Exam Insight — Most-Tested Formulas by Exam

CBSE: ε = −NdΦ/dt, L = μ₀N²A/l (derivation needed), U = ½LI², Transformer equation
NEET: ε = BLv, q = NΔΦ/R, Lenz's Law direction, Energy density u_B = B²/2μ₀
JEE Main: ε = ½BωL² (rotating rod), Rail problem full analysis (F = B²L²v/R), LC circuits
JEE Adv: Non-uniform B integration, inductors in series with mutual coupling (±2M), combined EMI + dynamics

❌ Most Common Formula Mistakes

1. Using θ as angle with surface instead of normal in Φ = BAcosθ
2. Forgetting N (number of turns) in ε = −NdΦ/dt
3. Confusing L and M — both in Henry but different physical meaning
4. Using ε = BLv when v is not perpendicular to B (should use sinα)
5. Forgetting the −ve sign in self-induction EMF calculations (direction-based problems)

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