🧮 Formula & Dimensional Analysis
Master all formulas with derivations, dimensions, and when to use which one.
Force & Motion Formulas
Scalar form: F = qvB sin θ (when E = 0)
When to use: Calculating force on moving charged particle in electromagnetic field
Units: F [N], q [C], v [m/s], B [T]
Forgetting sin θ when v and B are not perpendicular. Always check angle!
Derivation: qvB = mv²/r → r = mv/(qB)
When to use: Particle moving perpendicular to uniform magnetic field
Key insight: r ∝ p (momentum). Heavier/faster → larger radius
Critical: Independent of velocity! This is why cyclotron works.
"Prove time period is independent of speed" - derive T = 2πr/v and substitute r = mv/(qB)
When to use: Charged particle enters magnetic field at angle θ
Remember: v∥ causes forward motion, v⊥ causes circular motion
Vector form: F⃗ = I(L⃗ × B⃗)
Derivation: F = (nAL) × (evdB) = BIL
Direction: Fleming's Left Hand Rule
Same direction: Attractive
Opposite direction: Repulsive
Definition of Ampere: I₁ = I₂ = 1A, d = 1m → F/L = 2×10-7 N/m
Magnetic Field Formulas
μ₀ = 4π × 10-7 T·m/A
When to use: Finite wires, arcs, any irregular geometry
Direction: Right-hand thumb rule (curl fingers around wire)
Most asked in: NEET, JEE Main (direct substitution)
N = number of turns, R = radius
Direction: Perpendicular to coil plane
Writing R² in denominator. It's R, not R²!
At center (x = 0): B = μ₀NI/(2R)
Far away (x >> R): B ≈ (μ₀NIR²)/(2x³) = (μ₀M)/(2πx³)
θ in radians!
Full circle (θ = 2π): B = μ₀I/(2R) ✓
Semicircle (θ = π): B = μ₀I/(4R)
n = N/L = turns per unit length
Inside: Uniform field parallel to axis
Outside: ≈ 0 for ideal solenoid
Derive using Ampere's law with rectangular loop. Show Boutside = 0
N = total turns, r = distance from center
Field exists only inside toroid
Core: B = 0
Outside: B = 0
When to use: High symmetry (solenoid, toroid, infinite wire)
Biot-Savart: Any geometry, specific point
Ampere: High symmetry, easier calculation
Applications Formulas
Maximum torque (θ = 90°): τmax = NIAB
Minimum torque (θ = 0°): τ = 0 (equilibrium)
Unit: A·m² or J/T
Direction: Perpendicular to coil plane (right-hand rule)
θ = 0°: U = -MB (stable, minimum)
θ = 90°: U = 0
θ = 180°: U = +MB (unstable, maximum)
Work done: W = MB(cos θ₁ - cos θ₂)
Current Sensitivity: Is = θ/I = NAB/k
Voltage Sensitivity: Vs = θ/V = NAB/(kR)
Conversion to Ammeter:
Conversion to Voltmeter:
R = radius of dee
Key: Frequency independent of velocity
Dimensional Analysis
Key Quantities & Dimensions
| Quantity | Symbol | Unit | Dimension |
|---|---|---|---|
| Magnetic Field | B | Tesla (T) or Wb/m² | [M L⁰ T⁻² A⁻¹] |
| Magnetic Flux | Φ | Weber (Wb) | [M L² T⁻² A⁻¹] |
| Permeability | μ₀ | T·m/A or H/m | [M L T⁻² A⁻²] |
| Magnetic Dipole Moment | M | A·m² or J/T | [L² A] |
| Force | F | Newton (N) | [M L T⁻²] |
LHS: [f] = [T⁻¹]
RHS: [qB/m] = [A·T] × [M L⁰ T⁻² A⁻¹] / [M]
= [A × M L⁰ T⁻² A⁻¹ × M⁻¹] = [T⁻¹] ✓
Dimensional Check Practice
Problem: Verify r = mv/(qB) dimensionally
Solution:
[r] = [M × L T⁻¹] / [A × M L⁰ T⁻² A⁻¹]
= [M L T⁻¹] / [M T⁻² A × A⁻¹]
= [M L T⁻¹] / [M T⁻²]
= [L T⁻¹ × T²] = [L T] / [T] = [L] ✓