⚡ Current Electricity
formula sheet📑 Quick Navigation
PART A electric current & ohm's law
1Electric Current
$$I = \frac{Q}{t} = neAv_d$$
\(I\) = Electric current (A) |
\(Q\) = Charge (C) |
\(t\) = Time (s) |
\(n\) = Number density of electrons |
\(e\) = Charge of electron |
\(A\) = Area of cross-section |
\(v_d\) = Drift velocity
TRICK Current is the rate of flow of charge. Direction: from +ve to -ve terminal (conventional current).
2Drift Velocity
$$v_d = \frac{eE\tau}{m}$$
\(\tau\) = Relaxation time |
\(m\) = Mass of electron |
\(E\) = Electric field
$$v_d = \frac{I}{neA}$$
TRICK Drift velocity is very small (\(\approx 10^{-4}\) m/s).
3Mobility
$$\mu = \frac{v_d}{E} = \frac{e\tau}{m}$$
\(\mu\) = Mobility (m²/V·s)
TRICK Mobility = Drift velocity per unit electric field.
4Ohm's Law
$$V = IR$$
\(V\) = Potential difference (V) |
\(I\) = Current (A) |
\(R\) = Resistance (Ω)
TRICK Ohm's Law is valid for ohmic conductors (linear V-I graph).
5Resistance, Resistivity & Conductivity
$$R = \frac{V}{I} = \rho \frac{l}{A}$$
\(\rho\) = Resistivity (Ω·m) |
\(l\) = Length of conductor (m) |
\(A\) = Area of cross-section (m²)
$$\sigma = \frac{1}{\rho} = \frac{ne^2\tau}{m}$$
\(\sigma\) = Conductivity (S/m) |
\(n\) = Electron density
TRICK \(R \propto l\) and \(R \propto 1/A\). Resistivity depends only on material and temperature.
6Temperature Dependence of Resistance
$$\rho_T = \rho_0 [1 + \alpha (T - T_0)]$$
\(\alpha\) = Temperature coefficient of resistance (°C⁻¹) |
\(\rho_0\) = Resistivity at reference temperature
$$R_T = R_0 [1 + \alpha (T - T_0)]$$
TRICK For metals: \(\alpha > 0\) (resistance increases with temperature). For semiconductors: \(\alpha < 0\).
PART B resistance & grouping
7Series Combination of Resistances
$$R_{\text{eq}} = R_1 + R_2 + R_3 + \cdots$$
\(R_{\text{eq}}\) = Equivalent resistance (Ω)
TRICK In series: \(I\) same, \(V\) divides. \(R_{\text{eq}} >\) largest individual resistance.
8Parallel Combination of Resistances
$$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots$$
TRICK In parallel: \(V\) same, \(I\) divides. \(R_{\text{eq}} <\) smallest individual resistance.
9Mixed Grouping of Resistances
Combination of series and parallel — simplify step by step using series and parallel formulas.
TRICK Start from the farthest branch and simplify towards the main circuit.
PART C cells & combinations
10EMF and Internal Resistance
$$\mathcal{E} = V + Ir$$
\(\mathcal{E}\) = EMF of cell (V) |
\(V\) = Terminal potential difference |
\(I\) = Current |
\(r\) = Internal resistance (Ω)
$$V = \mathcal{E} - Ir$$
TRICK When \(I = 0\) (open circuit), \(V = \mathcal{E}\). When \(R \to 0\) (short circuit), \(I = \mathcal{E}/r\).
11Cells in Series
$$\mathcal{E}_{\text{eq}} = \mathcal{E}_1 + \mathcal{E}_2 + \mathcal{E}_3 + \cdots$$
$$r_{\text{eq}} = r_1 + r_2 + r_3 + \cdots$$
TRICK For identical cells: \(\mathcal{E}_{\text{eq}} = n\mathcal{E}\), \(r_{\text{eq}} = nr\).
12Cells in Parallel
$$\mathcal{E}_{\text{eq}} = \frac{\mathcal{E}_1/r_1 + \mathcal{E}_2/r_2 + \mathcal{E}_3/r_3 + \cdots}{1/r_1 + 1/r_2 + 1/r_3 + \cdots}$$
$$\frac{1}{r_{\text{eq}}} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \cdots$$
TRICK For identical cells: \(\mathcal{E}_{\text{eq}} = \mathcal{E}\), \(r_{\text{eq}} = r/n\).
13Parallel Combination of Two Cells of Unequal EMFs
$$\mathcal{E}_{\text{eq}} = \frac{\mathcal{E}_1 r_2 + \mathcal{E}_2 r_1}{r_1 + r_2}$$
$$r_{\text{eq}} = \frac{r_1 r_2}{r_1 + r_2}$$
TRICK Current through external resistance \(R\): \(I = \dfrac{\mathcal{E}_{\text{eq}}}{R + r_{\text{eq}}}\).
14Series Combination of n Cells of Unequal EMFs
$$\mathcal{E}_{\text{eq}} = \sum_{i=1}^{n} \mathcal{E}_i$$
$$r_{\text{eq}} = \sum_{i=1}^{n} r_i$$
15Mixed Grouping of Cells
If \(m\) rows in parallel, each row has \(n\) cells in series:
$$\mathcal{E}_{\text{eq}} = n\mathcal{E}$$
$$r_{\text{eq}} = \frac{nr}{m}$$
$$I_{\text{max}} = \frac{mn\mathcal{E}}{mR + nr}$$
TRICK For maximum current: \(R = \dfrac{nr}{m}\).
PART D kirchhoff's laws · bridges · potentiometer
16Kirchhoff's Laws
KCL (Junction Rule):
$$\sum I_{\text{in}} = \sum I_{\text{out}}$$
KVL (Loop Rule):
$$\sum \mathcal{E} = \sum IR$$
TRICK KCL: Charge conservation. KVL: Energy conservation (in a closed loop).
17Wheatstone Bridge
$$\frac{P}{Q} = \frac{R}{S}$$
\(P, Q, R, S\) = Four resistances of the bridge
$$\text{Balanced: } I_g = 0$$
TRICK At balance, galvanometer shows zero deflection. Used to find unknown resistance.
18Metre Bridge (Slide Wire Bridge)
$$\frac{P}{Q} = \frac{l}{100 - l}$$
\(l\) = Balancing length (cm)
$$P = Q \frac{l}{100 - l}$$
TRICK Metre bridge is a practical application of Wheatstone bridge.
19Potentiometer — Principle
$$V = \frac{IR}{L} \times l = k l$$
\(k\) = Potential gradient (V/m) |
\(l\) = Balancing length (m) |
\(L\) = Total length of wire |
\(I\) = Current in potentiometer wire
$$k = \frac{V}{L} = \frac{I R}{L}$$
TRICK Potentiometer is a zero deflection instrument. It measures potential difference accurately.
20Potentiometer — Comparing EMFs of Two Cells
$$\frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{l_1}{l_2}$$
\(l_1, l_2\) = Balancing lengths for cells 1 and 2
21Potentiometer — Measurement of Internal Resistance
$$r = R \left(\frac{l_1 - l_2}{l_2}\right)$$
\(R\) = Known external resistance |
\(l_1\) = Balancing length without shunt |
\(l_2\) = Balancing length with shunt
TRICK Compare EMFs: \(\mathcal{E}_1/\mathcal{E}_2 = l_1/l_2\). Internal resistance: \(r = R(l_1 - l_2)/l_2\).
📐 Units & Dimensions
| Quantity | SI Unit | Alternative / Equivalent | Dimension |
|---|---|---|---|
| Current \(I\) | A (ampere) | C/s | \(A\) |
| Charge \(Q\) | C (coulomb) | A·s | \(T\,A\) |
| Voltage \(V\) | V (volt) | J/C, W/A | \(M\,L^2\,T^{-3}\,A^{-1}\) |
| Resistance \(R\) | Ω (ohm) | V/A | \(M\,L^2\,T^{-3}\,A^{-2}\) |
| Resistivity \(\rho\) | Ω·m | — | \(M\,L^3\,T^{-3}\,A^{-2}\) |
| Conductivity \(\sigma\) | S/m (siemens/m) | 1/(Ω·m) | \(M^{-1}\,L^{-3}\,T^3\,A^2\) |
| Drift Velocity \(v_d\) | m/s | — | \(L\,T^{-1}\) |
| Mobility \(\mu\) | m²/(V·s) | — | \(M^{-1}\,L^0\,T^2\,A\) |
| EMF \(\mathcal{E}\) | V (volt) | J/C | \(M\,L^2\,T^{-3}\,A^{-1}\) |
| Internal Resistance \(r\) | Ω (ohm) | — | \(M\,L^2\,T^{-3}\,A^{-2}\) |
⚡ Quick Recall
| Current | \(I = \dfrac{Q}{t} = neAv_d\) |
| Drift Velocity | \(v_d = \dfrac{eE\tau}{m} = \dfrac{I}{neA}\) |
| Mobility | \(\mu = \dfrac{v_d}{E} = \dfrac{e\tau}{m}\) |
| Ohm's Law | \(V = IR\) |
| Resistance | \(R = \rho\dfrac{l}{A}\) |
| Conductivity | \(\sigma = \dfrac{1}{\rho} = \dfrac{ne^2\tau}{m}\) |
| Temp. Dependence | \(R_T = R_0[1 + \alpha(T - T_0)]\) |
| Series (Resistance) | \(R_{\text{eq}} = R_1 + R_2 + \cdots\) |
| Parallel (Resistance) | \(\dfrac{1}{R_{\text{eq}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots\) |
| EMF & Internal R | \(\mathcal{E} = V + Ir\) |
| Cells in Series | \(\mathcal{E}_{\text{eq}} = n\mathcal{E}, \quad r_{\text{eq}} = nr\) |
| Cells in Parallel | \(\mathcal{E}_{\text{eq}} = \mathcal{E}, \quad r_{\text{eq}} = \dfrac{r}{n}\) |
| Kirchhoff's KCL | \(\sum I_{\text{in}} = \sum I_{\text{out}}\) |
| Kirchhoff's KVL | \(\sum \mathcal{E} = \sum IR\) |
| Wheatstone Bridge | \(\dfrac{P}{Q} = \dfrac{R}{S}\) |
| Metre Bridge | \(P = Q \dfrac{l}{100 - l}\) |
| Potentiometer (Compare EMFs) | \(\dfrac{\mathcal{E}_1}{\mathcal{E}_2} = \dfrac{l_1}{l_2}\) |
| Potentiometer (Internal R) | \(r = R \dfrac{l_1 - l_2}{l_2}\) |
📘 Current Electricity · complete formula sheet with symbol meanings · updated 10 Aug 2026
Hi Please, do not Spam in Comments.