1. Electric Current
CBSENEETJEE
I = dQ/dt = nAev_d
SI Unit: Ampere (A) = C/s
Dimension: [A] = [A] (base quantity)
Nature: Scalar (but has direction by convention)
Variables: n=carrier density, A=area, e=charge, v_d=drift velocity
2. Drift Velocity
NEETJEE
v_d = eEτ/m = eVτ/(mL)
SI Unit: m/s
Dimension: [M⁰L¹T⁻¹]
Typical value: ~10⁻⁴ m/s
τ: relaxation time (~10⁻¹⁴ s)
3. Ohm's Law
CBSENEETJEE
V = IR
Resistance unit: Ohm (Ω) = V/A
Dimension of R: [M¹L²T⁻³A⁻²]
Valid for: Ohmic conductors at constant T
Slope of V-I graph: = R (constant for ohmic)
4. Resistance & Resistivity
CBSENEETJEE
R = ρL/A | ρ = m/(ne²τ)
ρ unit: Ω·m
Dimension of ρ: [M¹L³T⁻³A⁻²]
σ (conductivity): = 1/ρ, unit: S/m or (Ω·m)⁻¹
Dimension of σ: [M⁻¹L⁻³T³A²]
5. Temperature Coefficient
CBSENEET
R_T = R₀(1 + αΔT) | α = ΔR/(R₀ΔT)
α unit: per °C or per K (K⁻¹)
For metals: α > 0 (resistance increases)
For semiconductors: α < 0 (resistance decreases)
Dimension of α: [M⁰L⁰T⁰θ⁻¹]
6. Series Resistance
CBSENEETJEE
R_s = R₁ + R₂ + R₃ + ...
Voltage divider: V₁/V₂ = R₁/R₂ | Current same in all | R_s > any individual R
7. Parallel Resistance
CBSENEETJEE
1/R_p = 1/R₁ + 1/R₂ + ... | R_p = R₁R₂/(R₁+R₂) [2 only]
Current divider: I₁/I₂ = R₂/R₁ | Voltage same across all | R_p < smallest R
8. EMF & Terminal Voltage
CBSENEETJEE
V = ε - Ir (discharge) | V = ε + Ir (charge)
I (circuit): I = ε/(R+r)
Max power to R: when R = r (P_max = ε²/4r)
9. Cells in Series
CBSENEETJEE
ε_eff = nε | r_eff = nr | I = nε/(R+nr)
n identical cells in series. Best when R >> r.
10. Cells in Parallel
CBSENEETJEE
ε_eff = ε | r_eff = r/n | I = ε/(R+r/n)
n identical cells in parallel. Best when R << r.
11. Mixed Cell Grid (m×n)
NEETJEE
I = mnε/(mR+nr) | Max current when R = nr/m
m cells in series per row, n rows in parallel. Optimal: m/n = √(r/R)
12. KCL (Junction Law)
CBSENEETJEE
ΣI = 0 at any junction
Conservation of charge. ΣI_in = ΣI_out. Applies at every node.
13. KVL (Loop Law)
CBSENEETJEE
ΣV = 0 (closed loop)
Conservation of energy. Sum of potential differences = 0 around any loop.
14. Wheatstone Bridge Balance
CBSENEETJEE
P/Q = R/S (at balance, I_g = 0)
No current through galvanometer at balance. Most sensitive when P=Q=R=S.
15. Meter Bridge
CBSENEET
R/S = l/(100-l) → S = R(100-l)/l
l = balancing length in cm from left end. Wire of 100cm uniform cross-section.
16. Potentiometer — EMF Comparison
CBSENEETJEE
ε₁/ε₂ = l₁/l₂
Potential gradient k = V/L (V/m). V = kl. Balance when no current through galvanometer.
17. Potentiometer — Internal Resistance
CBSENEET
r = R(l₁/l₂ − 1) = R(l₁ − l₂)/l₂
l₁ = open circuit balance length | l₂ = balance length with R connected
18. Joule's Heating Law
CBSENEETJEE
H = I²Rt = VIt = V²t/R
Unit: Joule (J)
Dimension: [M¹L²T⁻²]
19. Electrical Power
CBSENEETJEE
P = VI = I²R = V²/R
Unit: Watt (W) = J/s
Dimension: [M¹L²T⁻³]
Max power to R: R = r (P_max = ε²/4r)
Efficiency: η = R/(R+r)
20. Current Density & Vector Ohm's Law
JEE
J = I/A = σE (vector form) | J = nev_d
J unit: A/m²
J dimension: [A¹L⁻²]
σ: conductivity (S/m)
J is vector; I is scalar
Dimensional Analysis
| Quantity | Symbol | SI Unit | Dimensional Formula | Exam Tag |
|---|---|---|---|---|
| Electric Current | I | Ampere (A) | [A] | CBSE |
| Charge | Q | Coulomb (C) | [A·T] | CBSE |
| Potential Difference | V | Volt (V) | [M¹L²T⁻³A⁻¹] | NEET |
| Resistance | R | Ohm (Ω) | [M¹L²T⁻³A⁻²] | JEE |
| Resistivity | ρ | Ω·m | [M¹L³T⁻³A⁻²] | JEE |
| Conductivity | σ | S/m | [M⁻¹L⁻³T³A²] | JEE |
| Current Density | J | A/m² | [A¹L⁻²] | JEE |
| Drift Velocity | v_d | m/s | [M⁰L¹T⁻¹] | NEET |
| Mobility | μ | m²/(V·s) | [M⁻¹L⁰T²A¹] | JEE |
| EMF | ε | Volt (V) | [M¹L²T⁻³A⁻¹] | CBSE |
| Power | P | Watt (W) | [M¹L²T⁻³] | CBSE |
| Energy (Heat) | H | Joule (J) | [M¹L²T⁻²] | CBSE |
| Temp. Coefficient (α) | α | K⁻¹ or °C⁻¹ | [M⁰L⁰T⁰θ⁻¹] | NEET |
| Relaxation Time | τ | second (s) | [M⁰L⁰T¹] | JEE |
JEE Main Dimensional Analysis Pattern
JEE Main regularly asks to find the dimension of quantities like mobility (μ = v_d/E), or to verify dimensional consistency of formulae like ρ = m/(ne²τ). The most tested: dimension of resistance and resistivity. Know [M¹L²T⁻³A⁻²] and [M¹L³T⁻³A⁻²] cold.
Dimension Error — Most Common
Students often write dimension of voltage as [ML²T⁻²] (energy dimension). Wrong! Voltage = Energy/Charge. So [V] = [ML²T⁻²]/[AT] = [ML²T⁻³A⁻¹]. Always work from definition.
Current
I = dQ/dt = nAev_d
A | [A] | I is conventionally opposite to electron flow
Drift Velocity
v_d = eEτ/m | v_d = I/(nAe)
m/s | Order ~10⁻⁴ m/s in copper wire
Ohm's Law
V = IR | R = V/I
Ω | Slope of V-I is R. Linear graph = ohmic.
Resistance
R = ρL/A | ρ = m/(ne²τ)
Ω·m for ρ | R ∝ L, R ∝ 1/A, R ∝ ρ
Temperature
R_T = R₀(1 + αΔT)
α = +ve (metals), α = −ve (semiconductors, thermistors)
Mobility
μ = v_d/E = eτ/m | σ = neμ
m²V⁻¹s⁻¹ | Relates conductivity to carrier properties
Series R
R_s = R₁ + R₂ + ... + Rₙ
V_total = V₁+V₂+..., I = same, R_s > R_max
Parallel R
1/R_p = 1/R₁ + 1/R₂ + ... + 1/Rₙ
I_total = I₁+I₂+..., V = same, R_p < R_min
KCL
ΣI_in = ΣI_out (at junction)
Conservation of charge. Node equation.
KVL
ΣV = 0 (closed loop)
ΣEMFs = ΣIR (algebraic). Direction matters for sign.
Max Power Transfer
P_max = ε²/4r (when R = r)
Maximum power to external R when R equals internal resistance r.
Efficiency
η = R/(R+r) × 100%
At max power (R=r): η = 50%. Higher R → higher efficiency.
Power — 3 Forms
P = VI = I²R = V²/R
Watt (W). Use the form where 2 of {V,I,R} are given.
Heat (Joule's Law)
H = I²Rt = VIt = V²t/R
Joule (J). H = Pt. In calorie: H = I²Rt/4.2 cal.
Bulb Resistance
R = V²_rated/P_rated
Lower rated P → Higher R. 40W bulb has higher R than 100W bulb.
Series Bulbs
P ∝ R (same I) → Lower rated P glows brighter
40W bulb brighter than 100W in series. Counter-intuitive but marks-scoring.
Parallel Bulbs
P ∝ 1/R (same V) → Higher rated P glows brighter
Normal operation. 100W brighter than 40W in parallel (as expected).
Commercial Energy
1 kWh = 3.6 × 10⁶ J
Electric bill unit. "1 unit" = 1 kWh.
Formula Bank Strategy
Don't memorize all 47 blindly. Priority order: (1) V=IR, P=VI, H=I²Rt — direct marks. (2) KCL/KVL — circuit problems. (3) ε=E-Ir — terminal voltage. (4) Potentiometer ratio. (5) Dimensional formulas for JEE Main analysis questions.