Complete revision notes and formulas for Current Electricity (Class 12). Curated for JEE, NEET, AP Physics, SAT, and CUET. Tap any topic to open the live simulation and full PYQ set.
V = IR with animated electron flow whose density scales with I.
Ohm's law: V = IR — applies only to ohmic materials at constant temperature.
I is rate of charge flow; one ampere = one coulomb per second.
Power dissipated: P = VI = I²R = V²/R.
Linear V-I relation for ohmic devices.
Three equivalent forms.
Diodes, bulbs (with hot filament), and electrolytes are NOT ohmic.
R depends on temperature: R(T) = R₀(1 + αΔT) for metals.
Compare ohmic (linear), diode (exponential), bulb (super-linear) curves.
Ohmic device: V-I straight line through origin, slope = R.
Diode: exponential I = Is(e^(V/V_T) − 1) — non-linear, asymmetric in V sign.
Bulb filament: super-linear (R rises with T) — concave V-I curve.
V_T ≈ 26 mV at room temperature.
Slope of V-I curve gives differential resistance dV/dI.
JEE often asks to identify device type from a V-I graph.
R = ρL/A — proportional to length, inversely proportional to cross-section.
Resistivity ρ is material-dependent: copper ~1.7 × 10⁻⁸ Ω·m, nichrome ~1.1 × 10⁻⁶ Ω·m.
Stretching a wire to length nL keeps mass constant → A drops to A/n → R becomes n²R.
Geometric × material.
Be careful with stretching: ρ doesn't change but L and A both change.
Conductivity σ = 1/ρ.
Cu, Fe, W (positive α) vs Si (negative α). Linear plot.
Metals: R(T) = R₀(1 + αΔT). α > 0 — resistance increases with T.
Semiconductors and insulators: α < 0 — resistance DECREASES with T (more carriers thermally generated).
Superconductors: R = 0 below critical temperature T_c.
Valid for small ΔT.
More accurate for non-metals.
Cu α ≈ 4 × 10⁻³/K. Si α negative and large in magnitude.
Light bulbs draw a surge current at switch-on (cold filament has lower R).
R_t = ΣRᵢ. Same I. V splits in proportion to each R.
Same current I flows through all components.
Voltage divides: V_total = ΣV_i = I·ΣR_i.
R_eq = ΣR_i — total is sum.
Direct addition.
If one element opens (breaks), entire circuit stops.
Voltage divider rule: V_i = V_total · R_i / R_eq.
1/R_t = Σ1/Rᵢ. Same V. I splits inversely with R.
Same voltage V across all branches.
Currents add: I_total = ΣI_i.
1/R_eq = Σ1/R_i. R_eq is smaller than the smallest branch.
Reciprocal addition.
Special case.
If one branch opens, others continue working.
Used in home wiring — appliances in parallel get full V independently.
Balance condition P/Q = R/S. Galvanometer reads zero when balanced.
Four resistors in diamond configuration with battery and galvanometer.
Balance condition: P/Q = R/S — galvanometer reads zero, no current through bridge.
Used to measure unknown resistance precisely.
At balance, V across galv = 0.
Strain gauges, temperature sensors all use Wheatstone bridges.
Sensitivity is highest when all four arms are comparable in resistance.
Slide jockey along 1m wire; X = R·ℓ/(100−ℓ) at balance.
Wheatstone bridge with one arm replaced by a uniform 1m resistance wire.
Slide jockey to find balance point at length ℓ → X = R·ℓ/(100−ℓ).
More precise than Wheatstone for similar values; uniform-wire ratio is the trick.
ℓ in cm.
Best balance is around 40–60 cm — minimises % error.
Wire must be uniform; check by reversing X and R.
2-loop circuit solved live. Currents update with KCL+KVL system.
KCL: Σ_in I = Σ_out I at any junction (charge conservation).
KVL: Σ V = 0 around any closed loop (energy conservation).
Together: solve any DC circuit by writing one equation per loop.
Junction rule.
Loop rule.
Sign conventions: rising EMF (− to +) is +, drop across R in direction of I is +.
Number of independent KCL equations = junctions − 1; KVL = mesh count.
Two opposing EMFs around a loop — i = (ε₁−ε₂)/(R₁+R₂).
Single-loop with two opposing batteries: i = (ε₁ − ε₂)/(R₁+R₂).
Mesh currents are a tidy way to apply KVL when you have many loops.
Sign convention: choose direction of mesh current; if answer is negative, actual is reversed.
Net EMF / total R.
If two batteries are in series with same orientation, EMFs add.
Wires have negligible R unless told otherwise.
V = ε − Ir. As R drops, more voltage is wasted across r.
Real cell: emf ε, internal resistance r. Terminal voltage V = ε − Ir.
Power delivered to load = ε²R/(R+r)². Maximum when R = r (matched load).
When I = 0 (open circuit), V = ε.
Drops as I increases.
P_max = ε²/(4r).
Old/dead batteries have higher internal resistance.
Short-circuit current = ε/r — limits current even with R_load = 0.
y-intercept = ε, slope = −r. Short circuit at V=0.
Plot V vs I: y-intercept = ε, slope = −r.
x-intercept = ε/r = short-circuit current.
Used to measure r experimentally — fit a line and read slope.
Linear in I.
Slope is always negative for a real cell.
Larger r → steeper drop with I.
ε = kℓ. Slide jockey to find balance length on long wire.
Measures EMF without drawing current — the gold standard for cell EMF.
Set up potential gradient k = V_driver / L_wire; balance when galvanometer reads zero.
ε = k · ℓ where ℓ is balance length.
k = V_driver/L_wire.
Driver cell EMF must exceed test cell EMF.
More accurate than voltmeter because at balance no current flows through test cell.
H = I²Rt. Wire glows brighter at higher P. Live ΔT estimate.
H = I²Rt — heat dissipated when current I flows through R for time t.
Equivalent forms: H = V²t/R = VIt.
Underlies heaters, fuses, filament bulbs, and explains why thin wires get hot.
Heat in joules; t in seconds.
Fuses melt because of localised heating — designed to fail safely.
Power-rated devices: don't exceed rated current or they overheat.
P = VI = I²R = V²/R. Three bars stay equal as you change V, R.
All three forms equal: P = VI = I²R = V²/R. Pick the most convenient based on what's given.
Power consumed × time = energy used (kWh on the meter).
Bulbs rated by P at rated V — you can compute R = V²/P.
Use Ohm's law to switch between.
100 W bulb at 220 V: I = P/V = 0.45 A; R = V²/P = 484 Ω (when hot).
P scales as V² for ohmic loads — small voltage drop gives big power loss.
R₁ in series with R₂||R₃. Live currents in each branch.
Mix of series and parallel: simplify in stages, working from inside out.
Identify nodes; combine clearly parallel groups first, then series.
Total current I = ε / R_eq through the source.
Common JEE pattern.
Redraw the circuit if needed — symmetry can simplify dramatically.
Watch for short-circuited resistors (= 0 V across both ends → carry no current).
v_d = I/(nAe). Animated electrons with thermal motion + tiny drift.
Current = nAev_d — electrons move with average drift velocity v_d (typically mm/s).
Despite v_d being slow, signals propagate near speed of light because the EM field travels fast.
n (charge density), A (cross-section), e (electron charge) all enter linearly.
Microscopic origin of current.
For Cu: n ≈ 8.5 × 10²⁸/m³.
Mobility μ = v_d/E. σ = neμ.
σ = ne²τ/m. See ions + electrons + collisions on atomic scale.
Drude model: free electrons accelerate in E, scatter off ions every τ seconds.
Average drift v_d = eEτ/m. Conductivity σ = ne²τ/m.
Resistivity ρ = 1/σ. Increases with T because τ shortens.
Microscopic origin.
Linear in E.
τ for Cu at room T ≈ 2.5 × 10⁻¹⁴ s — extremely short.
Mean free path ℓ = v_F·τ where v_F is Fermi velocity (not v_d).
Series multiplies ε, parallel divides r. Toggle and watch I update.
Series: ε_eq = Σε_i, r_eq = Σr_i. Higher voltage but higher internal R.
Parallel (identical cells): ε_eq = ε, r_eq = r/n. Same V, lower internal R, more current.
Mixed grouping: choose to maximize current through given load.
n cells in series.
n identical cells in parallel.
For best current through R: number of cells in series so that nr ≈ R.
Don't mix cells of different EMFs in parallel — circulating current will flow.
R_load → 0. I = ε/r. Wire overheats — visualized with red glow.
When R_load → 0, current spikes to I_short = ε/r.
All energy is dissipated in r → wire heats rapidly, can melt or catch fire.
Why fuses, MCBs, and circuit breakers exist.
Limited only by internal R.
Maximum heating in cell.
Practical batteries can deliver very high short-circuit currents — dangerous.
Modern Li-ion has built-in protection circuits to prevent shorts.
Current Electricity on sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs). Free physics revision for Class 12, JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT Subject Physics, and CUET-UG.