3.1 Current and Potential Difference
Electric Current ($I$) is the rate of flow of electric charge ($Q$) through a conductor.
1. Formula for Electric Current
$$I = \frac{Q}{t}$$
- $I$ = Electric current ($\text{A}$ or $\text{C s}^{-1}$)
- $Q$ = Electric charge ($\text{C}$)
- $t$ = Time ($\text{s}$)
For electrons carrying elementary charge $e = 1.6 \times 10^{-19}\text{ C}$, total charge $Q = ne$, where $n$ is the number of electrons.
2. Potential Difference ($V$)
Potential Difference ($V$) between two points in a circuit is defined as the work done ($W$) or energy transferred ($E$) in moving one coulomb of charge from one point to the other.
$$V = \frac{W}{Q} = \frac{E}{Q}$$
- $V$ = Potential difference ($\text{V}$ or $\text{J C}^{-1}$)
- $W$ = Work done or energy converted ($\text{J}$)
- $Q$ = Charge ($\text{C}$)
3.2 Resistance
Resistance ($R$) is the measure of the opposition to the flow of electric current in a conductor.
1. Ohm's Law
Ohm's Law states that the electric current ($I$) flowing through an ohmic conductor is directly proportional to the potential difference ($V$) across it, provided physical conditions (such as temperature) remain constant.
$$V = IR \implies R = \frac{V}{I}$$
2. Factors Affecting Resistance of a Wire
The resistance of a uniform conductor is given by the formula:
$$R = \rho \frac{l}{A}$$
- Length ($l$): $R \propto l$. Longer wire $\implies$ Higher resistance.
- Cross-Sectional Area ($A$): $R \propto \frac{1}{A}$. Thicker wire $\implies$ Lower resistance.
- Resistivity ($\rho$): Intrinsic property of the material ($\Omega\text{ m}$). Materials with lower $\rho$ (e.g., copper, silver) are better conductors.
- Temperature ($T$): Resistance of metallic conductors increases with rising temperature due to higher lattice ion vibration.
3.3 Electromotive Force (e.m.f.) and Internal Resistance
1. Electromotive Force ($\mathcal{E}$) vs. Terminal Potential Difference ($V$)
- Electromotive Force ($\mathcal{E}$): Total energy provided by a source (like a battery) to drive one coulomb of charge around a complete closed circuit ($\text{V}$ or $\text{J C}^{-1}$). Measured when the circuit is open ($I = 0$).
- Terminal Potential Difference ($V$): Work done to move one coulomb of charge across the external circuit components ($R$). Measured when the circuit is closed ($I > 0$).
2. Internal Resistance ($r$)
Internal resistance ($r$) is the resistance against the movement of charge carriers inside the chemical cell or power source itself.
$$ \mathcal{E} = V + Ir = I(R + r) $$
$$\text{Lost Volts} = Ir = \mathcal{E} - V$$
3. Determining $\mathcal{E}$ and $r$ from $V$ vs $I$ Graph
Rearranging $V = \mathcal{E} - Ir$ into a linear equation $y = mx + c$:
- $y$-intercept = Electromotive force ($\mathcal{E}$)
- Gradient ($m$) = Negative internal resistance ($-r$)
3.4 Electrical Energy and Power
1. Electrical Energy ($E$) Formulae
$$E = V I t = I^2 R t = \frac{V^2}{R} t$$
2. Electrical Power ($P$) Formulae
Power is the rate at which electrical energy is dissipated or converted into other forms:
$$P = \frac{E}{t} = V I = I^2 R = \frac{V^2}{R}$$
3. Energy Consumption Calculation
Electrical energy consumption is commercially measured in kilowatt-hours ($\text{kWh}$), where $1\text{ kWh} = 1\text{ unit}$ of electricity.
$$\text{Energy (kWh)} = \text{Power (kW)} \times \text{Time (hours)}$$
$$1\text{ kWh} = (1000\text{ W}) \times (3600\text{ s}) = 3.6 \times 10^6\text{ J}$$
4. Efficiency of Electrical Appliances
$$\text{Efficiency} = \frac{\text{Useful Output Power}}{\text{Input Power}} \times 100\%$$