Current Electricity deals with moving charges and their effects. This topic forms the foundation for electrical circuits and electronics.
Overview
graph TD
A[Current Electricity] --> B[Basic Concepts]
A --> C[Ohm's Law]
A --> D[Combinations]
A --> E[Kirchhoff's Laws]
A --> F[Instruments]
D --> D1[Series]
D --> D2[Parallel]
F --> F1[Wheatstone Bridge]
F --> F2[Potentiometer]Electric Current
Rate of flow of charge:
$$\boxed{I = \frac{dQ}{dt} = \frac{Q}{t}}$$Unit: Ampere (A) = Coulomb/second
Drift Velocity
$$v_d = \frac{eE\tau}{m}$$Current in terms of drift velocity:
$$\boxed{I = nAv_d e}$$where:
- n = number density of electrons
- A = cross-sectional area
- e = electron charge
Current Density
$$J = \frac{I}{A} = nev_d$$Mobility
$$\mu = \frac{v_d}{E} = \frac{e\tau}{m}$$Relation: $\sigma = ne\mu$
Ohm’s Law
$$\boxed{V = IR}$$Resistance
$$R = \frac{\rho L}{A}$$where:
- ρ = resistivity
- L = length
- A = cross-sectional area
Conductivity
$$\sigma = \frac{1}{\rho}$$Temperature Dependence
$$R_T = R_0(1 + \alpha\Delta T)$$where α = temperature coefficient of resistance
| Material | α behavior |
|---|---|
| Metals | Positive (R increases with T) |
| Semiconductors | Negative (R decreases with T) |
| Alloys | Very small |
Electrical Power and Energy
Power:
$$P = VI = I^2R = \frac{V^2}{R}$$Energy:
$$E = Pt = VIt = I^2Rt = \frac{V^2t}{R}$$Heat produced (Joule’s Law):
$$H = I^2Rt$$Combination of Resistors
Series Combination
$$\boxed{R_{eq} = R_1 + R_2 + R_3 + ...}$$- Same current through all resistors
- Voltage divides proportionally
Voltage divider:
$$V_1 = \frac{R_1}{R_1 + R_2}V$$Parallel Combination
$$\boxed{\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...}$$- Same voltage across all resistors
- Current divides inversely
Current divider:
$$I_1 = \frac{R_2}{R_1 + R_2}I$$Special Cases
Two resistors in parallel:
$$R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$$n equal resistors:
- Series: $R_{eq} = nR$
- Parallel: $R_{eq} = \frac{R}{n}$
EMF and Internal Resistance
Terminal voltage:
$$V = \varepsilon - Ir$$where:
- ε = EMF of cell
- r = internal resistance
Cells in Series
$$\varepsilon_{eq} = \varepsilon_1 + \varepsilon_2$$ $$r_{eq} = r_1 + r_2$$Cells in Parallel
$$\varepsilon_{eq} = \frac{\varepsilon_1/r_1 + \varepsilon_2/r_2}{1/r_1 + 1/r_2}$$ $$\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2}$$For identical cells (ε, r) in parallel:
$$\varepsilon_{eq} = \varepsilon, \quad r_{eq} = \frac{r}{n}$$Kirchhoff’s Laws
Junction Rule (KCL)
Sum of currents entering = Sum of currents leaving
$$\sum I_{in} = \sum I_{out}$$Loop Rule (KVL)
Sum of potential differences around any closed loop = 0
$$\sum V = 0$$graph LR
A[Kirchhoff's Laws] --> B[KCL: Conservation of charge]
A --> C[KVL: Conservation of energy]Wheatstone Bridge
Balanced condition:
$$\boxed{\frac{P}{Q} = \frac{R}{S}}$$When balanced, no current flows through galvanometer.
Metre Bridge
Special case of Wheatstone bridge:
$$\frac{R}{S} = \frac{l}{100-l}$$Potentiometer
Principle: Potential drop ∝ Length
$$\frac{V_1}{V_2} = \frac{l_1}{l_2}$$Applications
Comparison of EMFs:
$$\frac{\varepsilon_1}{\varepsilon_2} = \frac{l_1}{l_2}$$Internal resistance:
$$r = R\left(\frac{l_1 - l_2}{l_2}\right)$$Advantages over voltmeter:
- Draws no current at balance
- More accurate
- Can measure EMF directly
Galvanometer Conversion
To Ammeter (low resistance)
Add shunt resistance $S$ in parallel:
$$S = \frac{I_g G}{I - I_g}$$To Voltmeter (high resistance)
Add high resistance $R$ in series:
$$R = \frac{V}{I_g} - G$$Practice Problems
A wire of resistance 10Ω is stretched to double its length. Find new resistance.
Find equivalent resistance between A and B when three 6Ω resistors are connected as a triangle.
In a potentiometer, a cell of EMF 2V balances at 60 cm. Another cell balances at 80 cm. Find its EMF.
A galvanometer of resistance 50Ω gives full deflection for 1 mA. Convert it to (a) ammeter of range 10 A (b) voltmeter of range 10 V.