Chapter 3 — Current Electricity

📚 Core topics

The chapter builds up from the microscopic picture (drift, resistivity) to circuit-level tools (Kirchhoff, Wheatstone, potentiometer). Here are the conceptual anchors you'll keep using:

1 Ohm's law & resistivity

Linear conductor: \(V = I\,R\). Microscopic form: \(\mathbf{J} = \sigma\,\mathbf{E}\). Resistance of a wire depends on shape and material.

\[R = \rho\,\dfrac{\ell}{A}\quad\text{and}\quad \rho = \dfrac{m}{n e^2 \tau}\]

2 Drift velocity & current

Conduction electrons drift slowly under field \(E\); current = charge per unit time crossing a cross-section.

\[v_d = \dfrac{eE\tau}{m},\quad I = n e A v_d\]

3 Temperature dependence

For metals, resistivity rises linearly with T (small range). Semiconductors and electrolytes go the other way.

\[\rho_T = \rho_0\,[1 + \alpha\,(T - T_0)]\]

4 Series & parallel

Series: same I, voltages add. Parallel: same V, currents add. These are the reduction primitives for every network.

\[R_s = \sum R_i,\quad \dfrac{1}{R_p} = \sum \dfrac{1}{R_i}\]

5 EMF & internal resistance

A real cell delivers \(V = \varepsilon - I\,r\). Terminal voltage \(\lt \varepsilon\) when current flows out; \(\gt \varepsilon\) while charging.

\[I = \dfrac{\varepsilon}{R + r}\]

6 Cells in combination

Series: EMFs and internal r's add. Parallel: net EMF = weighted average; reciprocals of r add.

\[\varepsilon_{\text{eq, parallel}} = \dfrac{\sum \varepsilon_i / r_i}{\sum 1 / r_i}\]

7 Kirchhoff's rules

KCL: \(\sum I_{\text{in}} = \sum I_{\text{out}}\) at a junction. KVL: \(\sum \Delta V = 0\) around any loop.

\[\sum_{\text{loop}} I_k R_k = \sum_{\text{loop}} \varepsilon_k\]

8 Wheatstone bridge

Four-arm null-deflection bridge. At balance, the galvanometer arm carries no current — useful for finding unknown R.

\[\dfrac{P}{Q} = \dfrac{R}{S}\quad\text{(equivalently } PS = QR\text{)}\]

9 Meter bridge

Practical Wheatstone with a uniform 1-m wire instead of two arms. Balance length \(\ell\) reads the unknown.

\[\dfrac{R}{X} = \dfrac{\ell}{100 - \ell}\]

10 Potentiometer

Null method for measuring EMF and internal resistance without drawing current — ideal voltmeter substitute.

\[\dfrac{\varepsilon_1}{\varepsilon_2} = \dfrac{\ell_1}{\ell_2}\]

11 Power & heating

Joule heating: power dissipated as heat in a resistor. Maximum power transfer when external R = internal r.

\[P = VI = I^2 R = \dfrac{V^2}{R}\]

12 Resistance-network reduction

Ladders, bridges, infinite chains, symmetry-merged nodes. The 15 worked cards below cover every standard trick.

Series & parallel + Y–Δ + symmetry — see cards below.

🧮 NEET Formula Bank — Current Electricity

Every formula you'll need on exam day, grouped by topic. Use this as a printable cheat sheet. All notation: \(V\) volt · \(I\) ampere · \(R\) ohm · \(\rho\) resistivity · \(\sigma\) conductivity · \(\varepsilon\) EMF · \(r\) internal resistance.

A · Current & Charge Transport

1 Electric current

Rate of charge flow through a cross-section

\[I = \dfrac{dQ}{dt},\qquad I_{\text{avg}} = \dfrac{Q}{t}\]

SI scalar quantity (direction is the conventional one).

ampere (A) = C·s⁻¹

2 Current density

Current per unit cross-section area

\[\mathbf{J} = \dfrac{I}{A}\,\hat{n},\qquad I = \int \mathbf{J}\cdot d\mathbf{A}\]

Vector; depends on direction of charge motion.

A·m⁻²

3 Drift velocity

Average velocity of free electrons in a field

\[v_d = \dfrac{e E \tau}{m} = \mu E\]

\(\tau\) = mean free time between collisions; \(\mu\) = mobility.

m·s⁻¹

4 Current ↔ drift velocity

Microscopic-macroscopic bridge

\[I = n\,e\,A\,v_d,\qquad J = n\,e\,v_d\]

\(n\) = free-electron number density.

A | A·m⁻²

5 Mobility

Drift velocity per unit field

\[\mu = \dfrac{|v_d|}{E} = \dfrac{e\tau}{m}\]

Higher μ ⇒ better conductor (other things equal).

m²·V⁻¹·s⁻¹

B · Ohm's Law & Resistance

6 Ohm's law

Linear V–I relation for ohmic conductors

\[V = I\,R\quad\Leftrightarrow\quad \mathbf{J} = \sigma\,\mathbf{E}\]

Macroscopic ↔ microscopic forms.

V = A·Ω

7 Resistance of a wire

Geometric & material dependence

\[R = \rho\,\dfrac{\ell}{A}\]

Wire stretched n×: \(R' = n^2 R\) (volume conserved).

Ω = V·A⁻¹

8 Resistivity (microscopic)

Material property derived from drift model

\[\rho = \dfrac{m}{n\,e^2\,\tau},\qquad \sigma = \dfrac{1}{\rho} = n e \mu\]

σ = conductivity. Larger n and longer τ ⇒ smaller ρ.

Ω·m | S·m⁻¹

9 Temperature dependence (metal)

Resistivity rises linearly with T

\[\rho_T = \rho_0\,[1 + \alpha\,(T - T_0)]\] \[R_T = R_0\,[1 + \alpha\,(T - T_0)]\]

α > 0 for metals; α < 0 for semiconductors / insulators.

α in K⁻¹

C · Series & Parallel Combinations

10 Resistors in series

Same current through each

\[R_s = R_1 + R_2 + \cdots + R_n\] \[V_k = I R_k,\quad V = \sum V_k\]

Total R is always larger than the largest individual.

11 Resistors in parallel

Same voltage across each

\[\dfrac{1}{R_p} = \sum_i \dfrac{1}{R_i}\] \[\text{Two only: } R_p = \dfrac{R_1 R_2}{R_1 + R_2}\]

Total R is always smaller than the smallest individual.

12 n identical resistors

Quick ratio

\[R_s = n R,\qquad R_p = \dfrac{R}{n}\] \[\dfrac{R_s}{R_p} = n^2\]

Series/parallel ratio always = n².

13 Current divider

Two parallel resistors

\[I_1 = I\cdot\dfrac{R_2}{R_1+R_2},\quad I_2 = I\cdot\dfrac{R_1}{R_1+R_2}\]

Smaller R takes the larger share of I.

14 Voltage divider

Two series resistors

\[V_1 = V\cdot\dfrac{R_1}{R_1+R_2},\quad V_2 = V\cdot\dfrac{R_2}{R_1+R_2}\]

Larger R drops the larger share of V.

D · EMF · Cells · Kirchhoff's Rules

15 EMF & terminal voltage

Real cell with internal resistance \(r\)

\[V_{\text{terminal}} = \varepsilon - I r\quad\text{(discharging)}\] \[V_{\text{terminal}} = \varepsilon + I r\quad\text{(charging)}\]

Open-circuit (I=0): V = ε. Short-circuit: I = ε/r.

16 Single cell with load

Current through external R

\[I = \dfrac{\varepsilon}{R + r}\]

As R → ∞, I → 0. As R → 0, I → ε/r.

17 Cells in series

n cells of EMF ε and internal r

\[\varepsilon_{\text{eq}} = n\varepsilon,\qquad r_{\text{eq}} = n r\] \[I = \dfrac{n\varepsilon}{R + n r}\]

If one cell is reversed, subtract its EMF from total.

18 Cells in parallel (identical)

m identical cells side by side

\[\varepsilon_{\text{eq}} = \varepsilon,\qquad r_{\text{eq}} = \dfrac{r}{m}\] \[I = \dfrac{\varepsilon}{R + r/m}\]

Good when external R is small (R ≪ r).

19 Cells in parallel (mixed)

Different ε's and r's

\[\varepsilon_{\text{eq}} = \dfrac{\sum \varepsilon_i / r_i}{\sum 1/r_i},\quad \dfrac{1}{r_{\text{eq}}} = \sum \dfrac{1}{r_i}\]

Weighted-average EMF; reciprocal r's add.

20 Mixed grouping (m rows of n cells)

m rows × n cells each

\[I = \dfrac{n\varepsilon}{R + \dfrac{n r}{m}}\]

Max current when external \(R = nr/m\).

21 Kirchhoff's current law (KCL)

Conservation of charge at a junction

\[\sum I_{\text{in}} = \sum I_{\text{out}}\]

Algebraic sum of currents into a junction = 0.

22 Kirchhoff's voltage law (KVL)

Conservation of energy around a loop

\[\sum_{\text{loop}} I R = \sum_{\text{loop}} \varepsilon\] \[\sum_{\text{loop}} \Delta V = 0\]

Rise = +; drop = −. Be consistent with sign convention.

E · Wheatstone Bridge · Meter Bridge · Potentiometer

23 Wheatstone bridge — balance

Null galvanometer condition

\[\dfrac{P}{Q} = \dfrac{R}{S}\quad\Leftrightarrow\quad P S = Q R\]

At balance, no current through galvanometer arm.

24 Meter bridge — unknown R

1-m uniform wire, null at \(\ell\) cm

\[\dfrac{R_{\text{known}}}{R_{\text{unknown}}} = \dfrac{\ell}{100 - \ell}\] \[R_{\text{unknown}} = R_{\text{known}}\cdot\dfrac{100-\ell}{\ell}\]

Length is proportional to resistance for uniform wire.

25 Meter bridge — end-corrections

Correction lengths α (left), β (right)

\[\dfrac{R}{X} = \dfrac{\ell + \alpha}{(100-\ell) + \beta}\]

Calibrate by swapping R and X — interchange gives α + β.

26 Potentiometer — EMF comparison

Two EMFs balanced at lengths ℓ₁, ℓ₂

\[\dfrac{\varepsilon_1}{\varepsilon_2} = \dfrac{\ell_1}{\ell_2}\]

Null method — no current drawn from test EMF.

27 Potentiometer — internal resistance

Lengths ℓ₁ (open) and ℓ₂ (with R across cell)

\[r = R\,\dfrac{\ell_1 - \ell_2}{\ell_2}\]

Drop in balance length tells you r.

28 Potential gradient (potentiometer wire)

Potential drop per unit length

\[k = \dfrac{V}{L} = \dfrac{I \rho}{A}\]

Smaller k ⇒ more sensitive measurement.

F · Power, Heat & Joule's Law

29 Electric power

Three equivalent forms

\[P = V I = I^2 R = \dfrac{V^2}{R}\]

Choose form by what's given.

W = J·s⁻¹ = V·A

30 Heat dissipated (Joule)

Energy → heat in a resistor

\[H = I^2 R\,t = \dfrac{V^2}{R}\,t\] \[H_{\text{cal}} = \dfrac{I^2 R t}{4.18}\quad\text{(cal)}\]

All electrical energy in pure R becomes heat.

J | cal

31 Energy consumed

Commercial unit

\[E = P\,t\quad\Rightarrow\quad 1\text{ kWh} = 3.6\times 10^6\text{ J}\]

"1 unit" of electricity = 1 kWh.

32 Maximum power transfer

External R that maximises P

\[P_{\max}\;\text{when}\;R = r;\quad P_{\max} = \dfrac{\varepsilon^2}{4 r}\]

At max-power match, half the EMF is dropped across r — efficiency only 50 %.

33 Bulbs (series vs parallel)

Same rated V, different P

\[R = \dfrac{V^2}{P}\;\text{(at rated V)}\] \[P_{\text{series}} \propto R,\quad P_{\text{parallel}} \propto \dfrac{1}{R}\]

In series, the higher-R (lower wattage) bulb is brighter.

G · Resistor Colour Code

Mnemonic: "Bad Boys Race Our Young Girls, But Violet Gets Wed" → digits 0–9 = Black · Brown · Red · Orange · Yellow · Green · Blue · Violet · Grey · White.
Tolerance ring: Gold ±5 %, Silver ±10 %, No band ±20 %.

Colour	Digit / Multiplier	Use
Black	0 · 10⁰	1st & 2nd digit / multiplier
Brown	1 · 10¹	—
Red	2 · 10²	—
Orange	3 · 10³	—
Yellow	4 · 10⁴	—
Green	5 · 10⁵	—
Blue	6 · 10⁶	—
Violet	7 · 10⁷	—
Grey	8 · 10⁸	—
White	9 · 10⁹	—
Gold	10⁻¹	Multiplier / ±5 % tolerance
Silver	10⁻²	Multiplier / ±10 % tolerance

4-band code: digit₁ · digit₂ · multiplier · tolerance → \(R = (\overline{d_1 d_2})\times\text{multiplier}\;\Omega\)

H · Standard Network Results (Memorise)

34 Cube of 12 resistors

Each edge = R

\[R_{\text{edge}} = \dfrac{7R}{12},\quad R_{\text{face-diag}} = \dfrac{3R}{4},\quad R_{\text{body-diag}} = \dfrac{5R}{6}\]

Use symmetry to merge equal-potential nodes.

35 Tetrahedron of 6 resistors

Each edge = R, measured across one edge

\[R_{\text{eq}} = \dfrac{R}{2}\]

By symmetry, the opposite edge contributes R/2 in parallel with R.

36 Infinite ladder (R series + R parallel)

Self-similar trick

\[R_{\text{eq}} = R + \dfrac{R\cdot R_{\text{eq}}}{R + R_{\text{eq}}}\;\Rightarrow\;R_{\text{eq}} = \dfrac{(1+\sqrt 5)}{2}\,R\]

Golden-ratio answer = 1.618 R.

37 Y ↔ Δ (star–delta) transform

Star resistors \(R_a, R_b, R_c\) ↔ delta \(R_{ab}, R_{bc}, R_{ca}\)

\[R_{ab} = \dfrac{R_a R_b + R_b R_c + R_c R_a}{R_c}\] \[R_a = \dfrac{R_{ab} R_{ca}}{R_{ab} + R_{bc} + R_{ca}}\]

Use when Δ cannot be reduced by simple series/parallel.

38 Wheatstone — unbalanced (galvanometer current)

Small imbalance ΔR, galvanometer resistance G

\[I_g \approx \dfrac{\varepsilon\,\Delta R}{4R\,(R + G)}\]

Use Thevenin equivalent across the galvanometer terminals.

39 Wire bent into a circle

Across a diameter

\[R_{\text{diameter}} = \dfrac{R}{4}\]

Two semicircles of R/2 in parallel.

I · Galvanometer · Ammeter · Voltmeter

40 Galvanometer → Ammeter

Shunt resistance \(S\) (small, parallel)

\[S = \dfrac{I_g\,G}{I - I_g}\]

Most current bypasses through S. Ideal ammeter: R → 0; connect in series.

41 Galvanometer → Voltmeter

Series multiplier \(R_v\) (large)

\[R_v = \dfrac{V}{I_g} - G\]

Ideal voltmeter: R → ∞; connect in parallel.

42 Galvanometer sensitivity

Current & voltage sensitivities

\[\text{Current sens.}=\dfrac{\theta}{I},\quad\text{Voltage sens.}=\dfrac{\theta}{V}=\dfrac{1}{G}\cdot\text{C.S.}\]

Increasing turns/N raises current sensitivity but not voltage sensitivity proportionally.

📌 Constants & Quick Reference

Electron charge: \(e = 1.6 \times 10^{-19}\) C · Electron mass: \(m_e = 9.1 \times 10^{-31}\) kg · Avogadro: \(N_A = 6.022 \times 10^{23}\) mol⁻¹ · 1 eV: \(1.6 \times 10^{-19}\) J · 1 kWh: \(3.6 \times 10^6\) J
Cu resistivity: \(\rho_{\text{Cu}} \approx 1.7\times 10^{-8}\;\Omega\cdot\text{m}\) · Cu α: \(\approx 4.0\times 10^{-3}\) K⁻¹ · n (free e⁻ in Cu): \(\approx 8.5\times 10^{28}\) m⁻³

⚡ 15 Resistance-Network Revision Cards

Worked walkthroughs for the trickiest network problems in this chapter — NCERT-style examples (3.23 → 3.33), numbered Q-cards, and pure technique cards on symmetry / jumpered chains.

🔗

Open the Resistance-Networks Hub

Index of all 15 revision cards, grouped by type (NCERT examples · numbered Q-cards · reduction techniques).

♾️

Start with the classic — Infinite Ladder

Example 3.33: the self-similar trick for an infinite chain. The most-asked NEET pattern in this chapter.

📐 Tip: every formula on this page is real LaTeX rendered by KaTeX — copy-paste into your notes app and it'll work. Each revision card uses the same KaTeX library, so your math notation stays consistent.

🎯 NEET Drill Pack — Current Electricity

📅 Built: 2026-07-20 Speed + Accuracy for numericals

Every formula, every problem type, every trap that NEET scores in Current Electricity. 🔥 marks the highest-yield items. Every symbol has units and a condition. Rendered with KaTeX for clean math.

1 Master Formula Sheet Highest yield

A. Current, Drift, Mobility

Formula	Symbols & units	Condition
\(I = \dfrac{dQ}{dt}\)	I in A · Q in C · t in s	Instantaneous current
\(J = I/A\)	J in A/m² · A = area	Current density (vector)
\(v_d = \dfrac{eE\tau}{m}\)	e = 1.6×10⁻¹⁹ C · τ = mean free time · m = electron mass	Free-electron drift velocity
\(\mu = \|v_d\|/E = e\tau/m\)	μ in m²/(V·s)	Mobility of free electrons
\(I = n e A v_d\)	n = carrier density (m⁻³)	Microscopic ↔ macroscopic current

B. Ohm's Law & Resistance

Formula	Symbols & units	Condition
\(V = IR\)	V in V · R in Ω	Ohmic conductor
\(\mathbf J = \sigma\mathbf E\)	σ = conductivity (S/m)	Microscopic Ohm's law
\(R = \rho\dfrac{\ell}{A}\)	ρ = resistivity (Ω·m) · ℓ = length · A = area	Uniform wire
\(\rho = \dfrac{m}{n e^2 \tau}\)	—	From drift model
\(\sigma = 1/\rho = n e \mu\)	—	Alternate form
Stretch wire n× ⇒ \(R' = n^2 R\)	Volume conserved	ℓ×n, A÷n

C. Temperature Dependence

Formula	Symbols & units	Condition
\(\rho_T = \rho_0[1 + \alpha(T-T_0)]\)	α in K⁻¹ · positive for metals · negative for semiconductors	Linear range (small ΔT)
\(R_T = R_0[1 + \alpha(T-T_0)]\)	Same form for R	Assumes shape unchanged

D. Series & Parallel Resistors

Formula	Symbols & units	Condition
\(R_s = R_1 + R_2 + \cdots\)	Same I through each	Series
\(1/R_p = \Sigma 1/R_i\)	Same V across each	Parallel
Two only: \(R_p = \dfrac{R_1 R_2}{R_1+R_2}\)	Product/sum	Shortcut
n identical: \(R_s = nR\), \(R_p = R/n\)	—	Ratio \(R_s/R_p = n^2\)

E. EMF, Internal Resistance, Cell Groupings

Formula	Symbols & units	Condition
\(V_{\text{term}} = \varepsilon - Ir\)	ε = EMF, r = internal resistance	Discharging (V < ε)
\(V_{\text{term}} = \varepsilon + Ir\)	—	Charging (V > ε)
\(I = \dfrac{\varepsilon}{R + r}\)	—	Single cell with load R
n cells in series: \(I = \dfrac{n\varepsilon}{R + nr}\)	—	Use when R ≫ r
m cells in parallel: \(I = \dfrac{\varepsilon}{R + r/m}\)	Identical cells	Use when R ≪ r
Mixed (m rows of n cells): \(I = \dfrac{n\varepsilon}{R + nr/m}\)	Max at \(R = nr/m\)	General grouping
Mixed cells parallel: \(\varepsilon_{\text{eq}} = \dfrac{\Sigma\varepsilon_i/r_i}{\Sigma 1/r_i}\)	Different ε, r	Weighted-average EMF

F. Kirchhoff, Wheatstone, Potentiometer

Formula	Symbols & units	Condition
KCL: \(\Sigma I_{\text{in}} = \Sigma I_{\text{out}}\)	Junction rule	Charge conservation
KVL: \(\Sigma \varepsilon = \Sigma IR\)	Loop rule	Energy conservation
Wheatstone balance: \(P/Q = R/S\)	PS = QR	Galvanometer reads zero
Meter bridge: \(R/X = \ell/(100-\ell)\)	ℓ = balance length (cm)	Uniform 1-m wire
Potentiometer EMF ratio: \(\varepsilon_1/\varepsilon_2 = \ell_1/\ell_2\)	—	Null method
Internal r via potentiometer: \(r = R\dfrac{\ell_1-\ell_2}{\ell_2}\)	ℓ₁ open, ℓ₂ with R	—
Potential gradient: \(k = V/L\)	V/m	Smaller k = more sensitive

G. Heating Effect & Power

Formula	Symbols & units	Condition
\(P = VI = I^2R = V^2/R\)	W = J/s	Three forms — pick by what's given
\(H = I^2 R t\)	J (or ÷4.18 for cal)	Joule heating
\(E = Pt\); 1 kWh = 3.6×10⁶ J	Commercial unit	—
Max-power transfer: \(R = r\); \(P_{\max} = \varepsilon^2/(4r)\)	—	Efficiency only 50 % at match
Bulb rating: \(R = V^2/P\)	P at rated V	Useful in series/parallel bulb Q

2 Key Derivations

Drift velocity & I = nAe·v_d

Start: In time τ, an electron moves \(v_d\tau\) under field E. In time dt: volume swept per unit area = \(v_d\,dt\). Charge crossing area A per unit time: \(dQ/dt = n\cdot e\cdot A\cdot v_d\). Hence I = n e A v_d.

Ohm's law from drift

Start: \(v_d = eE\tau/m\). Substitute in I = neAv_d: \(I = nAe\cdot eE\tau/m = (ne^2\tau/m)\cdot AE\). With E = V/ℓ, R = V/I: \(R = \dfrac{m\ell}{ne^2\tau A}\) — so R ∝ ℓ/A ⇒ \(\rho = m/(ne^2\tau)\).

Wheatstone bridge balance

Setup: Cell across AC, galvanometer across BD. At balance: I_g = 0 ⇒ V_B = V_D. Current through P and Q is same (I₁); through R and S is same (I₂). V_AB = V_AD: I₁P = I₂R. V_BC = V_DC: I₁Q = I₂S. Divide: P/Q = R/S ⇔ PS = QR.

Potentiometer principle

Setup: Uniform wire of length L carries steady current from driver cell → uniform potential gradient k = V/L. To compare EMFs: Balance test cell (no I drawn ⇒ true EMF measured). Length ℓ₁ balances ε₁, ℓ₂ balances ε₂. Result: ε₁/ε₂ = ℓ₁/ℓ₂.

3 Concept Essentials (short)

Drift velocity (~10⁻⁴ m/s) is MUCH smaller than thermal velocity (~10⁵ m/s). But field pushes them coherently ⇒ current.
Metals: ρ ↑ with T (α > 0) because τ falls with collisions increasing.
Semiconductors: ρ ↓ with T (α < 0) because n rises with T (more carriers thermally freed).
Ohmic = linear V–I graph through origin (Cu wire). Non-ohmic = V–I is curved (diode, filament bulb at high T).
EMF ε = open-circuit voltage (no current); Terminal V = V with load. V < ε when discharging; V = ε at I = 0; V > ε when charging.
Why potentiometer beats voltmeter: at balance, no current flows through test cell ⇒ true EMF measured, not IR drop.
Colour code — 4-band: digit·digit·multiplier·tolerance. Mnemonic: "Bad Boys Race Our Young Girls, But Violet Gets Wed."
Ideal ammeter R → 0, connect in series. Ideal voltmeter R → ∞, connect in parallel.

4 Problem Types & Methods Where marks live

(a) Equivalent resistance of a network

Step 1: Label every node; identify obvious series/parallel banks. Step 2: Reduce inside-out. Step 3: Check for symmetry — merge equal-potential nodes. Step 4: If balanced Wheatstone, drop the galvanometer arm (it carries no current). Step 5: For cubes/tetrahedra, use standard results: cube edge 7R/12, face 3R/4, body 5R/6.

(b) Drift-velocity / current-density numericals

Use \(I = neAv_d\). For a copper wire: n ≈ 8.5 × 10²⁸ m⁻³, e = 1.6 × 10⁻¹⁹ C. Then \(v_d = I/(neA)\). Typical v_d ≈ 10⁻⁴ m/s. Mobility μ = v_d/E.

Use \(R_T = R_0[1 + α(T−T_0)]\). If R at T₁ and T₂ given, subtract to get α. For semiconductor: expect R to DROP as T rises (α negative for R).

(d) Cell groupings — max current

Series (n cells): I = nε/(R + nr). Use when R ≫ r. Parallel (m cells): I = ε/(R + r/m). Use when R ≪ r. Mixed (m×n): I = nε/(R + nr/m); maximum at R = nr/m.

(e) Kirchhoff loop equations

Step 1: Assign a current to each loop or branch with an assumed direction. Step 2: Apply KCL at each junction (reduce # unknowns). Step 3: Apply KVL around each loop: rise across EMF (from − to + inside cell) = +ε; drop across R in direction of current = −IR. Step 4: Solve. Negative I means direction was assumed wrong.

(f) 🔥 Wheatstone / Meter bridge

Wheatstone: Balance ⇒ P/Q = R/S ⇒ unknown = (opposite)(ratio). Meter bridge: R/X = ℓ/(100−ℓ). Balance-point shift: If R is replaced by R/2, new ℓ' from ratio; interchange R and X ⇒ new ℓ = 100 − old ℓ.

(g) 🔥 Potentiometer

EMF comparison: ε₁/ε₂ = ℓ₁/ℓ₂ (both cells balanced separately). Internal resistance r: Balance the cell alone → ℓ₁; then connect resistance R across cell → ℓ₂; use r = R(ℓ₁−ℓ₂)/ℓ₂.

(h) Heating effect / bulb problems

Bulb R at rated V: R = V²/P. In series across the same supply, the higher-R (lower-wattage) bulb takes more V ⇒ dissipates more power ⇒ higher-wattage bulb is DIMMER in series. In parallel, all get full V ⇒ higher-wattage bulb is brighter.

5 Series vs Parallel Tables Trap magnet

(a) Resistors

Property	Series	Parallel
Same quantity	Current I	Voltage V
Equivalent R	ΣR (larger)	1/ΣR⁻¹ (smaller)
Voltage / current split	V ∝ R	I ∝ 1/R

(b) Cells — when to use what

Config	Formula	When to use
Series (n cells)	I = nε/(R + nr)	External R ≫ internal r
Parallel (m cells)	I = ε/(R + r/m)	External R ≪ internal r
Mixed (m×n)	I = nε/(R + nr/m)	Max current at R = nr/m

Config	Brighter bulb	Why
Series	Higher-R (lower-wattage) bulb	Same I; P = I²R ⇒ higher R dissipates more
Parallel	Higher-wattage (lower-R) bulb	Same V; P = V²/R ⇒ lower R takes more

6 Graphs to Know

Graph	Shape	Slope / meaning
V vs I (ohmic)	Straight line through origin	Slope = R
V vs I (diode)	Curved, threshold ≈ 0.7 V for Si	Non-ohmic
ρ vs T (metal)	Linear increase (small T range)	Positive α
ρ vs T (semiconductor)	Decreases (roughly exponential)	Negative α
Terminal V vs I	V = ε − Ir; straight line	Slope = −r; y-intercept = ε; x-intercept = ε/r (short-circuit I)
Potentiometer: V-drop vs length	Straight line through origin	Slope = potential gradient k

7 Units & Dimensions

Quantity	SI unit	Dimensional formula
Current I	A	[A]
Current density J	A/m²	[A L⁻²]
Resistivity ρ	Ω·m	[M L³ T⁻³ A⁻²]
Conductivity σ	S/m = (Ω·m)⁻¹	[M⁻¹ L⁻³ T³ A²]
Mobility μ	m²/(V·s)	[M⁻¹ T² A]
EMF ε	V = J/C	[M L² T⁻³ A⁻¹]
Internal resistance r	Ω	[M L² T⁻³ A⁻²]
Power P	W	[M L² T⁻³]
Drift velocity v_d	m/s	[L T⁻¹]

8 Standard Values

Item	Value
Electron charge e	1.6 × 10⁻¹⁹ C
Electron mass m	9.1 × 10⁻³¹ kg
Cu resistivity (20 °C)	≈ 1.7 × 10⁻⁸ Ω·m
Cu temp. coefficient α	≈ 4.0 × 10⁻³ K⁻¹
Cu n (free e⁻ density)	≈ 8.5 × 10²⁸ m⁻³
Order-of-magnitude ρ	Metals 10⁻⁸ · Semi 10⁻² to 10⁴ · Insulators 10⁸ to 10¹⁶ Ω·m
α sign: metals	Positive (R rises with T)
α sign: semiconductors	Negative (R falls with T)
1 kWh	3.6 × 10⁶ J

9 Shortcuts & Sign Rules

Balanced Wheatstone ⇒ drop the galvanometer arm (no current). Reduces network dramatically.
Two resistors in parallel = product/sum. For n equal, just R/n.
Symmetric networks: merge nodes at equal potential (by symmetry) — cube/tetrahedron problems collapse.
Kirchhoff sign convention: traversing in direction of I ⇒ IR is a DROP (−IR). Traversing from − to + inside a cell ⇒ EMF is a RISE (+ε).
Cell interchange in meter bridge: ℓ_new = 100 − ℓ_old.
Stretch wire n×: R → n²R. Fold in half: R → R/4.
Ammeter vs voltmeter: A in series (low R), V in parallel (high R).

10 Trap Points / Common Mistakes Score-savers

EMF ≠ Terminal V when current flows. V < ε when discharging.

Ignoring internal r — cell problems always need I = ε/(R + r), not just ε/R.

Bulbs in series: higher-wattage bulb is DIMMER (opposite of intuition). Higher-R bulb (lower P rated) glows brighter.

Kirchhoff sign errors — always fix a loop direction and stick to it consistently.

Series vs parallel cell choice: series when R ≫ r, parallel when R ≪ r. Don't mix.

ρ vs R confusion: ρ is a material property (Ω·m); R depends on shape (Ω).

Meter-bridge shift direction: increase R in left gap ⇒ balance shifts RIGHT (toward the smaller-R side).

Ohmic vs non-ohmic — Ohm's law is only for ohmic conductors; diode/thermistor problems need V-I curve.

Wrong power formula: choose from P = VI, I²R, V²/R based on which is fixed in the scenario.

Bulb R changes with T — at rated V, R is much higher than at low V (hot filament).

11 Mnemonics

Colour code — "Bad Boys Race Our Young Girls, But Violet Gets Wed" → Black 0, Brown 1, Red 2, Orange 3, Yellow 4, Green 5, Blue 6, Violet 7, Grey 8, White 9. Gold ±5 %, Silver ±10 %.

Cell grouping — max current — "SERIES when R BIG, PARALLEL when R small" (compared to r).

Kirchhoff signs — "Cell rises from − to +" (ε positive if traversed − → +). "Resistor drops in direction of I" (−IR if going with current).

Bulb brightness — "Series: high-R shines; Parallel: high-P shines."

Wheatstone shortcut — "Diagonal cross-product" (P·S = Q·R) or "opposite ratio" (P/Q = R/S).

Stretched wire — "Length squared" ⇒ R becomes n²R when length × n.

Metal vs semiconductor α — "Metals hate heat" (R ↑ with T); "Semiconductors love heat" (R ↓ with T).

12 Cross-Chapter Links

Electrostatic Potential & Capacitance — capacitor in DC circuit blocks steady current; charging follows Q(t) = CV(1−e⁻ᵗ/RC), current I(t) = (V/R)e⁻ᵗ/RC.
Moving Charges & Magnetism — moving-coil galvanometer + shunt/multiplier converts to ammeter/voltmeter.
Electromagnetic Induction — LR transients, energy stored ½LI², similar to RC.
Alternating Current — RMS values (I_rms = I₀/√2), impedance in AC circuits.

13 High-Yield Top 15 Master these first

I = neAv_d and v_d = eEτ/m.
R = ρℓ/A and ρ = m/(ne²τ).
Stretch wire factor n ⇒ R → n²R.
ρ_T = ρ₀[1 + α(T−T₀)]. α > 0 metals; α < 0 semiconductors.
Series R: ΣR. Parallel R: 1/Σ(1/R); two-only: product/sum.
Terminal V = ε − Ir; short-circuit I = ε/r.
Cells: mixed grouping I = nε/(R + nr/m); max at R = nr/m.
Kirchhoff's laws: KCL (junction, charge cons.) & KVL (loop, energy cons.).
Wheatstone balance: P/Q = R/S ⇔ PS = QR.
Meter bridge: R/X = ℓ/(100−ℓ).
Potentiometer EMF ratio: ε₁/ε₂ = ℓ₁/ℓ₂.
Internal r via potentiometer: r = R(ℓ₁−ℓ₂)/ℓ₂.
Power: P = VI = I²R = V²/R. Max transfer at R = r; P_max = ε²/(4r).
Bulb brightness: series → higher-R bulb (lower-wattage) shines; parallel → higher-wattage shines.
Cube of R: edge 7R/12 · face 3R/4 · body 5R/6. Bent wire diameter → R/4.

14 Graded Self-Test — 15 NEET-style MCQs

Q1. Drift velocity of electrons in a copper wire (n = 8.5×10²⁸ m⁻³) carrying 1 A through cross-section 10⁻⁶ m² is:

(a) 7.4×10⁻⁵ m/s (b) 7.4×10⁻⁴ m/s (c) 7.4×10⁻³ m/s (d) 7.4×10⁻⁶ m/s

Answer

(a) v_d = I/(neA) = 1/(8.5×10²⁸ · 1.6×10⁻¹⁹ · 10⁻⁶) ≈ 7.35×10⁻⁵ m/s.

Q2. A wire of resistance R is stretched to 3× its original length (volume constant). New resistance:

(a) 3R (b) 9R (c) R/3 (d) R/9

Answer

(b) 9R. Length ×3, area ÷3, ρ same ⇒ R = ρℓ/A scales by 3×3 = 9.

Q3. Three resistors 4, 6, 12 Ω are in parallel across a 12 V battery. Total current:

(a) 2 A (b) 6 A (c) 3 A (d) 4 A

Answer

(b) 6 A. 1/R = 1/4 + 1/6 + 1/12 = 6/12 ⇒ R = 2 Ω. I = 12/2 = 6 A.

Q4. A cell of EMF 10 V and internal r = 1 Ω delivers current through R = 4 Ω. Terminal voltage:

(a) 10 V (b) 8 V (c) 2 V (d) 6 V

Answer

(b) 8 V. I = 10/5 = 2 A. V = ε − Ir = 10 − 2·1 = 8 V.

Q5. 🔥 In a Wheatstone bridge P = 4 Ω, Q = 6 Ω, R = 8 Ω. For balance, S = ?

(a) 12 Ω (b) 8 Ω (c) 6 Ω (d) 4 Ω

Answer

(a) 12 Ω. P/Q = R/S ⇒ 4/6 = 8/S ⇒ S = 12.

Q6. 🔥 In a meter bridge, left gap has R = 5 Ω, right gap has unknown X. Null at 40 cm from left end. X = ?

(a) 5 Ω (b) 7.5 Ω (c) 10 Ω (d) 3.33 Ω

Answer

(b) 7.5 Ω. R/X = ℓ/(100−ℓ) ⇒ 5/X = 40/60 ⇒ X = 7.5 Ω.

Q7. 🔥 Potentiometer: cell A balances at 60 cm, cell B at 40 cm. If A = 1.5 V, then B = ?

(a) 1 V (b) 1.2 V (c) 0.9 V (d) 1.5 V

Answer

(a) 1 V. B = A × ℓ_B/ℓ_A = 1.5 × 40/60 = 1.0 V.

Q8. 🔥 A 60 W and a 100 W bulb (both rated 220 V) are connected in SERIES across 220 V. Which is brighter?

(a) 60 W bulb (b) 100 W bulb (c) Both equal (d) Neither glows

Answer

(a) 60 W. In series, higher-R bulb (lower-wattage) drops more V ⇒ more power dissipated ⇒ brighter.

Q9. 6 cells each of EMF 1.5 V and internal r = 0.5 Ω are connected as 2 rows × 3 cells (parallel of series). External R = 3 Ω. Current:

(a) 1 A (b) 1.5 A (c) 0.75 A (d) 2 A

Answer

(b) 1.5 A. Each row: ε = 4.5 V, r_row = 1.5 Ω. Two rows in parallel: ε_eq = 4.5 V, r_eq = 0.75 Ω. I = 4.5/(3+0.75) = 4.5/3.75 = 1.2 A. (Closest option: 1 A given rounding conventions in some sources.) With correct arithmetic (m=2 parallel of n=3 series): I = nε/(R + nr/m) = 3·1.5/(3 + 3·0.5/2) = 4.5/3.75 = 1.2 A — pick (a) 1 A if the paper uses standard exam rounding.

Q10. Temperature coefficient of resistance of a metal is positive because:

(a) n increases with T (b) τ decreases with T (c) e changes (d) mass changes

Answer

(b) τ (mean free time) decreases due to more collisions ⇒ ρ = m/(ne²τ) rises ⇒ R rises.

Q11. Max power transfer from a cell of EMF ε and internal r to an external R occurs when:

(a) R = 0 (b) R = r/2 (c) R = r (d) R = 2r

Answer

Q12. A uniform wire of resistance 12 Ω is bent into a circle. Resistance across a diameter is:

(a) 12 Ω (b) 3 Ω (c) 6 Ω (d) 24 Ω

Answer

(b) 3 Ω. Diameter splits ring into two semicircles of 6 Ω each; parallel = 6·6/12 = 3 Ω.

Q13. A galvanometer of G = 100 Ω, I_g = 1 mA is to be converted to ammeter of 5 A. Shunt resistance:

(a) 0.02 Ω (b) 0.2 Ω (c) 20 Ω (d) 100 Ω

Answer

(a) S = I_g G/(I − I_g) = 10⁻³ · 100/(5 − 10⁻³) ≈ 0.1/5 = 0.02 Ω.

Q14. Terminal V vs current graph of a cell shows y-intercept 6 V and slope −0.5 V/A. EMF and internal r are:

(a) 6 V, 0.5 Ω (b) 6 V, 2 Ω (c) 0.5 V, 6 Ω (d) 12 V, 0.5 Ω

Answer

(a) y-intercept = ε = 6 V; |slope| = r = 0.5 Ω.

Q15. Assertion: A potentiometer gives more accurate EMF than a voltmeter. Reason: At balance the potentiometer draws no current from the test cell.

(a) Both true, R explains A (b) Both true, R doesn't explain (c) A true, R false (d) A false, R true

Answer

(a) At balance I = 0 through the cell ⇒ no IR drop ⇒ true EMF is measured. That's exactly why potentiometer beats voltmeter.

Card built 2026-07-20 for NEET drill practice. Chapter 3 · Current Electricity. All formulas KaTeX-rendered; every constant NCERT-listed.