Formula & Dimensional Analysis
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Group 01
Basic Definitions & Capacitance
Capacitance
C = Q / V
Dimension: [M⁻¹L⁻²T⁴A²]
Parallel Plate (Air)
C = ε₀A / d
A = plate area, d = separation
Parallel Plate (Dielectric)
C = Kε₀A / d
K = dielectric constant ≥ 1
Spherical Capacitor
C = 4πε₀ab / (b-a)
a = inner, b = outer radius
Isolated Sphere
C = 4πε₀R
b → ∞, C = 4πε₀R
Cylindrical Capacitor
C = 2πε₀L / ln(b/a)
L = length, ln = natural log
Dielectric Constant
K = C / C₀ = ε / ε₀
Dimensionless ratio
Conducting Slab in Capacitor
C = ε₀A / (d - t)
t = thickness of conductor
Dielectric Slab (partial)
C = ε₀A / (d - t + t/K)
t = slab thickness, K = dielectric const
Polarization
P = ε₀(K-1)E = χₑε₀E
χₑ = K - 1 = susceptibility
Series — Equivalent
1/C = 1/C₁ + 1/C₂ + ...
C_eq < smallest C
Parallel — Equivalent
C = C₁ + C₂ + C₃ + ...
C_eq > largest C
Two Capacitors — Series
C = C₁C₂ / (C₁ + C₂)
Product over sum rule
Charge on Series Caps
Q₁ = Q₂ = Q (same charge)
For initially uncharged capacitors
Voltage Division (Series)
V₁/V₂ = C₂/C₁
Voltage inversely ∝ capacitance
Common Potential (Redistribution)
V = (C₁V₁+C₂V₂)/(C₁+C₂)
For same polarity connection
Energy in Capacitor (3 Forms)
U = ½CV² = ½QV = Q²/2C
All equivalent; use known quantities
Energy Density
u = ½ε₀E²
In dielectric: u = ½Kε₀E²
Energy Lost in Redistribution
ΔU = ½(C₁C₂)/(C₁+C₂)·(V₁-V₂)²
Always a loss (heat generated)
Time Constant
τ = RC
Unit: seconds. Dimension: [T]
Charging — Charge
q(t) = CV(1 - e^(-t/RC))
At t=τ: q = 0.632·CV
Discharging — Charge
q(t) = Q₀e^(-t/RC)
At t=τ: q = 0.368·Q₀
Half-life
t₁/₂ = 0.693·RC
= RC·ln2
Force Between Plates
F = Q²/(2ε₀A) = σ²A/(2ε₀)
Attractive force between opposite charges
Electric Field (Between Plates)
E = σ/ε₀ = V/d = Q/(ε₀A)
Uniform field; applies inside only
n Capacitors (identical) Series
C_eq = C/n
Each C in series
n Capacitors (identical) Parallel
C_eq = nC
Each C in parallel
Dimensional Analysis
Dimensions of Key Quantities
Dimensional analysis questions appear every year in JEE Main. Master these.
| Quantity | SI Unit | Dimensional Formula | Exam Relevance |
|---|---|---|---|
| Capacitance (C) | Farad (F) | [M⁻¹L⁻²T⁴A²] | High |
| Electric Field (E) | V/m = N/C | [MLT⁻³A⁻¹] | High |
| Permittivity (ε₀) | F/m = C²/Nm² | [M⁻¹L⁻³T⁴A²] | High |
| Energy Density (u) | J/m³ | [ML⁻¹T⁻²] | Medium |
| Electric Flux (Φ) | Vm = N·m²/C | [ML³T⁻³A⁻¹] | Medium |
| Charge (Q) | Coulomb (C) | [AT] | Basic |
| Potential (V) | Volt (V) | [ML²T⁻³A⁻¹] | High |
| Time Constant (RC) | Second (s) | [T] | Medium |
Dimensional Verification — Worked Example
Verify: C = ε₀A/d has correct dimensions
1
LHS: Capacitance C = [M⁻¹L⁻²T⁴A²]
2
RHS: ε₀A/d = [M⁻¹L⁻³T⁴A²] × [L²] / [L]
3
= [M⁻¹L⁻³⁺²⁻¹T⁴A²] = [M⁻¹L⁻²T⁴A²] ✓
⚡ The key: always reduce ε₀ to base dimensions first: [M⁻¹L⁻³T⁴A²]. This appears in JEE Main objective questions.
Formula Connections
How Formulas Relate
Energy Formula Derivation Chain
C = Q/V
→ V = Q/C
→ integrate dW = V·dq →
U = Q²/2C = ½CV² = ½QV
🧠 Key Insight
The three energy forms are NOT three different formulas — they're the same formula with substitutions: Q²/2C → replace Q = CV → ½CV². Replace V = Q/C → Q²/2C → ½QV. Understand this and you'll never get confused again.
Interactive Tool
Series/Parallel Calculator
⚡ Equivalent Capacitance Calculator