Formula Bank & Dimensional Analysis

Every formula you need. Searchable. Copy-paste ready. With dimensions and units.

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Coulomb's Law (Scalar)

F = k |q₁ q₂| / r²

Dimensions: [MLT⁻²]

Units: Newton (N)

Where: k = 9 × 10⁹ N·m²/C²

Coulomb's Law (Vector)

F⃗ = (k q₁ q₂ / r²) r̂

Dimensions: [MLT⁻²]

Direction: Along line joining charges

Electric Field

E = F/q = k Q/r²

Dimensions: [MLT⁻³A⁻¹]

Units: N/C or V/m

Electric Potential

V = k Q/r

Dimensions: [ML²T⁻³A⁻¹]

Units: Volt (V)

Potential Energy

U = k q₁q₂/r

Dimensions: [ML²T⁻²]

Units: Joule (J)

Work Done

W = q(V_B - V_A)

Dimensions: [ML²T⁻²]

Units: Joule (J)

Electric Dipole Moment

p = q × 2a

Dimensions: [LTA]

Units: C·m

Field on Axial Line (Dipole)

E = 2kp/r³

Valid for: r >> 2a

Field on Equatorial Line

E = kp/r³

Direction: Opposite to p⃗

Torque on Dipole

τ = p E sin θ

Vector form: τ⃗ = p⃗ × E⃗

Potential Energy (Dipole)

U = -p E cos θ = -p⃗·E⃗

Stable: θ = 0° (U = -pE)

Unstable: θ = 180° (U = +pE)

Linear Charge Density

λ = Q/L

Units: C/m

Surface Charge Density

σ = Q/A

Units: C/m²

Volume Charge Density

ρ = Q/V

Units: C/m³

Coulomb's Constant

k = 1/(4πε₀)

ε₀ = 8.85 × 10⁻¹² C²/(N·m²)

k = 9 × 10⁹ N·m²/C²

Charge Quantization

q = ±ne

e = 1.6 × 10⁻¹⁹ C

n = 1, 2, 3, ...

🎯 Formula Mastery Strategy

Don't memorize blindly. Understand:
1. Where the formula comes from (derivation)
2. When to use it (conditions/assumptions)
3. Common variations (vector vs scalar)
4. Units and dimensional checks

Dimensional Analysis Table

Quantity	Symbol	Dimensions	SI Unit
Electric Charge	q, Q	[AT]	Coulomb (C)
Electric Force	F	[MLT⁻²]	Newton (N)
Electric Field	E	[MLT⁻³A⁻¹]	N/C or V/m
Electric Potential	V	[ML²T⁻³A⁻¹]	Volt (V)
Electric Energy	U, W	[ML²T⁻²]	Joule (J)
Permittivity	ε₀	[M⁻¹L⁻³T⁴A²]	C²/(N·m²)
Dipole Moment	p	[LTA]	C·m
Torque	τ	[ML²T⁻²]	N·m

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