Current Electricity Formula Sheet
All key Current Electricity formulas for JEE Physics: current, Ohm's law, resistance, power, Kirchhoff's laws, Wheatstone & meter bridge. Quick revision for JEE Main & Advanced.
Every must-know formula and result from the Current Electricity chapter, grouped by sub-topic for last-minute revision. Use the tables to scan fast and the boxed formulas for the headline relations.
Electric Current and Drift Velocity
$$\boxed{I = \frac{dQ}{dt} = \frac{Q}{t}}$$$$\boxed{v_d = \frac{eE\tau}{m}} \qquad \boxed{I = nAev_d} \qquad v_d = \frac{I}{nAe}$$| Quantity | Formula | Notes |
|---|---|---|
| Electric current | $I = \dfrac{Q}{t}$ | Unit: ampere (A) = C/s |
| Drift velocity | $v_d = \dfrac{eE\tau}{m}$ | $\tau$ = relaxation time |
| Drift velocity (from $I$) | $v_d = \dfrac{I}{nAe}$ | Most useful JEE form |
| Current from drift | $I = nAev_d$ | Links micro to macro |
| Current density | $J = \dfrac{I}{A} = nev_d$ | Unit: A/m² |
| Current density (vector) | $\vec{J} = ne\vec{v_d}$ | Direction opposite to electron drift |
| Mobility | $\mu = \dfrac{v_d}{E} = \dfrac{e\tau}{m}$ | Unit: m²/(V·s) |
| Conductivity–mobility | $\sigma = ne\mu$ | |
| Current via mobility | $I = neA\mu E = neA\mu\dfrac{V}{L}$ | |
| Drift velocity (same V) | $v_d = \dfrac{V}{\rho L ne}$ | $v_d \propto 1/L$ for same V |
Microscopic Ohm’s law:
$$\boxed{\vec{J} = \sigma \vec{E}} \qquad \sigma = \frac{ne^2\tau}{m}$$Stretched wire (same current): $v_d \propto 1/A$, so if length doubles ($A$ halves), $v_d$ doubles.
Drift velocity is tiny ($\sim 10^{-4}$ m/s) but the signal speed $\approx c = 3\times10^8$ m/s — that is why the bulb lights instantly.
Copper number density: $n \approx 8.5 \times 10^{28}$ electrons/m³ (about one free electron per atom).
Ohm’s Law and V–I Characteristics
$$\boxed{V = IR}$$| Form | Expression | Use when |
|---|---|---|
| Current | $I = \dfrac{V}{R}$ | $V$, $R$ known |
| Resistance | $R = \dfrac{V}{I}$ | from V–I measurement |
| Microscopic | $J = \sigma E$ | vector / material form |
Units: Voltage = volt (J/C), Current = ampere (C/s), Resistance = ohm (V/A).
| Material type | V–I graph | Obeys Ohm’s law? |
|---|---|---|
| Metallic conductor (const. T) | Straight line through origin | Yes |
| Filament bulb | Curves (R rises with T) | No |
| Diode / LED | Threshold then steep rise | No |
| Electrolyte | Non-linear at low V | No (initially) |
V–I graph (V on y-axis): slope $= R$. I–V graph (I on y-axis): slope $= 1/R$ = conductance.
For a curved (non-ohmic) graph, $R = V/I$ at that point; dynamic resistance $= dV/dI$.
Ohm’s law is not universal — it holds only for ohmic conductors at constant temperature.
Resistance, Resistivity and Temperature
$$\boxed{R = \rho\frac{L}{A}} \qquad \boxed{R_T = R_0(1 + \alpha\Delta T)}$$| Quantity | Formula | Notes |
|---|---|---|
| Resistance | $R = \rho\dfrac{L}{A}$ | $R \propto L$, $R \propto 1/A$ |
| Resistivity | $\rho = R\dfrac{A}{L}$ | Material property (Ω·m) |
| Conductivity | $\sigma = \dfrac{1}{\rho}$ | Unit: S/m |
| Conductance | $G = \dfrac{1}{R} = \dfrac{I}{V} = \dfrac{\sigma A}{L}$ | Unit: siemens (S) |
| Temperature (metal) | $R_T = R_0[1+\alpha(T-T_0)]$ | $\alpha > 0$ for metals |
| Temperature (semicond.) | $R_T = R_0 e^{-\beta\Delta T} \approx R_0(1-\alpha\Delta T)$ | $\alpha < 0$ |
| Compare two wires | $\dfrac{R_1}{R_2} = \dfrac{\rho_1}{\rho_2}\cdot\dfrac{L_1}{L_2}\cdot\dfrac{A_2}{A_1}$ | ratio method |
| Hollow cylinder | $R = \dfrac{\rho L}{\pi(r_2^2 - r_1^2)}$ | $r_2$ outer, $r_1$ inner |
Increasing resistivity order: Silver < Copper < Gold < Aluminium (in $\rho$); copper $\rho \approx 1.7\times10^{-8}$ Ω·m.
| Material | $\alpha$ (per °C) | Behaviour |
|---|---|---|
| Metals (Cu, Ag, Al, W) | $\sim +4\times10^{-3}$ | R increases with T |
| Nichrome, Manganin | $\sim 10^{-4}$ or less | nearly constant |
| Semiconductors, electrolytes | Negative | R decreases with T |
Stretched to $n\times$ length (volume constant): $\boxed{R' = n^2 R}$.
Radius halved: length $\times 4$, area $\div 4$ ⟹ $R' = 16R$.
Cut into $n$ equal parts, all in parallel: $\boxed{R_{eq} = \dfrac{R}{n^2}}$.
Combination of Resistors
$$\boxed{R_{eq} = R_1 + R_2 + R_3 + \dots} \quad \text{(series)}$$$$\boxed{\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots} \quad \text{(parallel)}$$| Configuration | Result | Notes |
|---|---|---|
| Two in parallel | $R_{eq} = \dfrac{R_1 R_2}{R_1+R_2}$ | “product over sum” |
| $n$ equal in series | $R_{eq} = nR$ | same current in all |
| $n$ equal in parallel | $R_{eq} = \dfrac{R}{n}$ | same voltage across all |
| Voltage divider (series) | $V_1 = \dfrac{R_1}{R_1+R_2}V$ | voltage $\propto R$ |
| Current divider (parallel) | $I_1 = \dfrac{R_2}{R_1+R_2}I$ | current $\propto 1/R$ |
| Conductance in parallel | $G_{eq} = G_1 + G_2 + G_3$ | add directly |
EMF and Internal Resistance
$$\boxed{V = \varepsilon - Ir} \qquad I = \frac{\varepsilon}{R+r}$$| Configuration | EMF | Internal resistance |
|---|---|---|
| Cells in series | $\varepsilon_{eq} = \varepsilon_1 + \varepsilon_2$ | $r_{eq} = r_1 + r_2$ |
| Cells in parallel | $\varepsilon_{eq} = \dfrac{\varepsilon_1/r_1 + \varepsilon_2/r_2}{1/r_1 + 1/r_2}$ | $\dfrac{1}{r_{eq}} = \dfrac{1}{r_1}+\dfrac{1}{r_2}$ |
| $n$ identical cells parallel | $\varepsilon_{eq} = \varepsilon$ | $r_{eq} = \dfrac{r}{n}$ |
On discharge $V = \varepsilon - Ir$ (terminal V < EMF). Open circuit: $V = \varepsilon$. Short circuit: $I = \varepsilon/r$ (maximum current).
Electrical Power and Heating
$$\boxed{P = VI = I^2R = \frac{V^2}{R}}$$$$\boxed{H = I^2Rt = Pt}$$| Quantity | Formula | Use when |
|---|---|---|
| Power (basic) | $P = VI$ | $V$, $I$ given |
| Power (current form) | $P = I^2R$ | series — same $I$ |
| Power (voltage form) | $P = \dfrac{V^2}{R}$ | parallel — same $V$ |
| Heat (Joule’s law) | $H = I^2Rt = VIt = \dfrac{V^2t}{R}$ | energy in joules |
| Heat in calories | $H = \dfrac{I^2Rt}{4.2}$ | 1 cal = 4.2 J |
| Energy | $E = Pt = VIt = I^2Rt = \dfrac{V^2t}{R}$ | |
| Bulb resistance from rating | $R = \dfrac{V^2}{P}$ | $V$, $P$ = rated values |
| Commercial unit | $1\text{ kWh} = 3.6\times10^6$ J | “1 unit” of electricity |
Maximum power transfer: load $R$ = source internal resistance $r$.
$$\boxed{R = r} \qquad P_{max} = \frac{\varepsilon^2}{4r} \qquad \eta = 50\%$$Series (same $I$, use $P = I^2R$): higher R glows brighter (a 60 W bulb beats a 100 W bulb).
Parallel (same $V$, use $P = V^2/R$): lower R / higher rating glows brighter (100 W beats 60 W).
A “100 W, 220 V” rating holds only at rated voltage; at any other $V$ use $P = V^2/R$ with $R = V_{rated}^2/P_{rated}$ constant.
Heater coils — time $\propto R$ for fixed heat: series adds times ($t_1+t_2$), parallel adds reciprocals ($1/t = 1/t_1 + 1/t_2$).
Kirchhoff’s Laws
$$\boxed{\sum I_{in} = \sum I_{out}} \quad \text{(KCL — charge conservation)}$$$$\boxed{\sum V = 0} \quad \text{(KVL — energy conservation)}$$KVL alternative form: $\sum \varepsilon = \sum IR$.
| Element / traversal | Potential change | Notes |
|---|---|---|
| Resistor, with current | $-IR$ (drop) | current flows high → low V |
| Resistor, against current | $+IR$ (rise) | |
| Battery, $-$ to $+$ | $+\varepsilon$ (rise) | “uphill” |
| Battery, $+$ to $-$ | $-\varepsilon$ (drop) | “downhill” |
| Number of KCL equations | $(n-1)$ for $n$ junctions | nth is redundant |
| Number of KVL equations | one per independent loop | use mesh loops |
Traversing a resistor with the current is a potential drop ($-IR$); against it is a rise ($+IR$).
If a solved current comes out negative, the actual direction is opposite to the one you assumed.
Check: $\sum P_{\text{sources}} = \sum P_{\text{loads}}$ (power balance).
Wheatstone Bridge and Meter Bridge
$$\boxed{\frac{P}{Q} = \frac{R}{S}} \quad\Longleftrightarrow\quad \boxed{P\times S = Q\times R}$$At balance, no current flows through the galvanometer and $V_C = V_D$.
| Quantity | Formula | Notes |
|---|---|---|
| Balance condition | $\dfrac{P}{Q} = \dfrac{R}{S}$, i.e. $PS = QR$ | products of opposite arms equal |
| Unknown resistance | $R = \dfrac{P\times S}{Q}$ | three arms known |
| Balanced bridge $R_{eq}$ | $R_{eq} = \dfrac{(P+Q)(R+S)}{P+Q+R+S}$ | only when balanced |
| Meter bridge ($l$ from R side) | $R = P\times\dfrac{100-l}{l}$ | $l$ in cm on 100 cm wire |
| Meter bridge ($l$ from P side) | $R = P\times\dfrac{l}{100-l}$ | check which side $l$ is |
| Interchange gaps | $l_2 = 100 - l_1$ | complementary balance point |
| End corrections | $R = P\times\dfrac{100-l+\alpha}{l+\beta}$ | $\alpha,\beta$ end resistances |
| Shunt effect | $R' = \dfrac{R\,R_{shunt}}{R+R_{shunt}}$ | effective R decreases |
graph LR
A((A)) -- P --> C((C))
A -- R --> D((D))
C -- Q --> B((B))
D -- S --> B
C -- "G (galvanometer)" --- D
A -- "battery" --- BIf $P = R$, balance is at the centre, $l = 50$ cm. Best sensitivity is near $l = 50$ cm; poor near the ends.
Meter bridge wire is uniform, so resistance $\propto$ length — that is why the length ratio replaces the resistance ratio.
When balanced, current still flows through the P–Q and R–S branches; only the galvanometer branch carries none.
Galvanometer Conversion
$$\boxed{S = \frac{I_g G}{I - I_g}} \quad \text{(ammeter: shunt in parallel)}$$$$\boxed{R = \frac{V}{I_g} - G} \quad \text{(voltmeter: high R in series)}$$| Conversion | Added resistance | Position | Effect on circuit |
|---|---|---|---|
| To ammeter | low shunt $S = \dfrac{I_g G}{I - I_g}$ | parallel | low net resistance; connect in series |
| To voltmeter | high $R = \dfrac{V}{I_g} - G$ | series | high net resistance; connect in parallel |
Here $G$ = galvanometer resistance, $I_g$ = full-scale deflection current.
Potentiometer
Principle: potential drop $\propto$ length, so $\dfrac{V_1}{V_2} = \dfrac{l_1}{l_2}$.
| Application | Formula | Notes |
|---|---|---|
| Comparing EMFs | $\dfrac{\varepsilon_1}{\varepsilon_2} = \dfrac{l_1}{l_2}$ | balancing lengths |
| Internal resistance | $r = R\left(\dfrac{l_1 - l_2}{l_2}\right)$ | $l_1$ open, $l_2$ with $R$ across cell |
At balance it draws no current from the cell, so it measures true EMF — more accurate than a voltmeter, which always draws some current.
Constants and Standard Values
| Constant | Value |
|---|---|
| Electron charge $e$ | $1.6 \times 10^{-19}$ C |
| Electron mass $m$ | $9.1 \times 10^{-31}$ kg |
| Number density (copper) | $\approx 8.5 \times 10^{28}$ m⁻³ |
| Resistivity of copper | $\approx 1.7 \times 10^{-8}$ Ω·m |
| 1 kWh | $3.6 \times 10^{6}$ J |
| 1 calorie | $4.2$ J |