Electric Current and Circuits

🟢 Lite — Quick Review (1h–1d)

Rapid summary of electric current and circuits for NABTEB physics.

Electric current is the flow of electric charge, typically through a conductor. It is measured in Amperes (A).

Key Definitions:

• Electric current ($I$): Rate of flow of charge: $I = \frac{Q}{t}$ (1 A = 1 C/s)
• Potential difference ($V$): Work done per unit charge: $V = \frac{W}{Q}$ (1 V = 1 J/C)
• Resistance ($R$): Opposition to current flow: $R = \frac{V}{I}$ (1 Ω = 1 V/A)
• Electromotive force (emf): Energy source providing voltage (e.g., battery): $\varepsilon = \frac{W}{Q}$

Ohm’s Law: $$V = IR$$

This states that the potential difference across a conductor is directly proportional to the current flowing through it, provided temperature remains constant.

Resistivity: $$\rho = \frac{RA}{l}$$

Where $A$ is cross-sectional area, $l$ is length. Unit: Ω·m. Factors affecting resistance: length ($R \propto l$), cross-section ($R \propto 1/A$), material (resistivity $\rho$), temperature ($R$ increases with temperature for metals).

Power Dissipated: $$P = IV = I^2R = \frac{V^2}{R}$$

Unit: Watt (W). Energy: $E = Pt$.

Series and Parallel Circuits:

	Series	Parallel
Current	Same through all components	Divides at junctions
Voltage	Divides across components	Same across all components
Equivalent Resistance	$R_s = R_1 + R_2 + R_3$	$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

⚡ NABTEB Exam Tip: In parallel circuits, current divides. The larger resistance gets the smaller current share. Total resistance in parallel is ALWAYS less than the smallest individual resistance.

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🟡 Standard — Regular Study (2d–2mo)

For NABTEB students who want thorough understanding of electric circuits.

Series Circuits:

In a series circuit, all components are connected end-to-end, forming a single path for current.

Characteristics:

• Current is the same through all components: $I = I_1 = I_2 = I_3$
• Voltage divides: $V = V_1 + V_2 + V_3$
• Total resistance: $R_T = R_1 + R_2 + R_3$
• If one component fails (opens), the entire circuit stops working

Voltage Drop Across Each Resistor: $$V_1 = IR_1, \quad V_2 = IR_2, \quad V_3 = IR_3$$

Parallel Circuits:

In a parallel circuit, components are connected across the same two points, providing multiple paths for current.

Characteristics:

• Voltage is the same across all branches: $V = V_1 = V_2 = V_3$
• Current divides: $I = I_1 + I_2 + I_3$
• Total resistance: $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$
• If one branch opens, other branches continue working

Current Division in Two Parallel Resistors: $$I_1 = I \times \frac{R_2}{R_1 + R_2}, \quad I_2 = I \times \frac{R_1}{R_1 + R_2}$$

(More current flows through the smaller resistance)

EMF and Internal Resistance:

A real battery has internal resistance ($r$), which causes the terminal voltage to drop when current flows.

$$\varepsilon = V + Ir$$

Where $\varepsilon$ = emf, $V$ = terminal voltage, $I$ = current, $r$ = internal resistance.

Maximum Power Transfer: Power delivered to external load is maximum when $R_{\text{load}} = r$.

$$P_{\max} = \frac{\varepsilon^2}{4r}$$

Circuit Analysis — Solving Complex Circuits:

Step 1: Identify series and parallel combinations; simplify step by step Step 2: Calculate equivalent resistances Step 3: Use Ohm’s law to find total current Step 4: Work backwards to find currents and voltages in each branch

Kirchhoff’s Laws:

Kirchhoff’s Current Law (KCL): At any junction, the sum of currents entering equals the sum leaving. $$\sum I_{\text{in}} = \sum I_{\text{out}}$$

Kirchhoff’s Voltage Law (KVL): Around any closed loop, the sum of emfs equals the sum of potential drops. $$\sum \varepsilon = \sum IR$$

Heating Effect of Current:

When current flows through a resistor, electrical energy is converted to heat energy.

Joule’s Law: $$H = I^2Rt$$

Where $H$ = heat energy (in Joules, if $I$ is in A, $R$ in Ω, $t$ in s).

Applications:

• Electric kettle, iron, heater: use heating effect of current
• Fuse wires: melt when current exceeds safe limit, protecting circuits
• Filament bulbs: thin wire glows white-hot, producing light

⚡ NABTEB Exam Tip: For circuits with both series and parallel components, reduce the circuit step by step. Start from the innermost parallel branch, find its equivalent resistance, then combine with series resistors.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage of electric current and circuits for NABTEB students.

Detailed Derivation — Series and Parallel Resistors:

Series Derivation: Current $I$ flows through each resistor in succession. Voltage across each: $V_1 = IR_1$, $V_2 = IR_2$, $V_3 = IR_3$ Total voltage: $V = V_1 + V_2 + V_3 = I(R_1 + R_2 + R_3)$ Total resistance: $R_T = \frac{V}{I} = R_1 + R_2 + R_3$

Parallel Derivation: Each resistor is connected directly across the voltage source. Current through each: $I_1 = \frac{V}{R_1}$, $I_2 = \frac{V}{R_2}$, $I_3 = \frac{V}{R_3}$ Total current: $I = I_1 + I_2 + I_3 = V\left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\right)$ Total resistance: $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Special Case — Two Parallel Resistors: $$R_T = \frac{R_1 R_2}{R_1 + R_2}$$

Potential Divider:

A potential divider uses series resistors to obtain a fraction of the supply voltage: $$V_1 = V \times \frac{R_1}{R_1 + R_2}$$

Applications:

• Volume controls in audio equipment
• Setting reference voltages in circuits
• Biasing transistors

The Wheatstone Bridge:

A bridge circuit used to measure unknown resistance:

• When balanced: $\frac{R_1}{R_2} = \frac{R_3}{R_4}$
• Unknown resistance: $R_x = R_3 \times \frac{R_2}{R_1}$ (for the configuration where $R_1$ and $R_2$ are the ratio arms)

When NOT balanced, the bridge can be simplified using Kirchhoff’s laws.

Temperature Dependence of Resistance:

For metals: $R_T = R_0(1 + \alpha \Delta T)$ Where $\alpha$ = temperature coefficient of resistance (K⁻¹)

For semiconductors/thermistors: resistance DECREASES with increasing temperature (negative temperature coefficient).

For superconductors: resistance drops to zero below critical temperature.

Conductors, Insulators, and Semiconductors:

Material	Resistivity	Behaviour
Conductor	~10⁻⁸ to 10⁻⁶ Ω·m	Low resistance
Insulator	~10¹⁰ to 10²⁰ Ω·m	Very high resistance
Semiconductor	~10⁻⁵ to 10⁶ Ω·m	Conductivity between conductor and insulator

EMF vs Terminal Voltage:

For a battery with emf $\varepsilon$ and internal resistance $r$, supplying current $I$ to external resistance $R$:

• Internal potential drop: $Ir$
• Terminal voltage: $V = \varepsilon - Ir$
• When $I = 0$ (open circuit): $V = \varepsilon$ (maximum terminal voltage)
• When $r \gg R$ (short circuit approximation): $V \approx 0$

Measuring Instruments:

Ammeter:

• Connected IN SERIES with the circuit
• Should have very low resistance (ideal = 0 Ω) to minimise circuit disturbance
• Converts to show current in Amperes

Voltmeter:

• Connected IN PARALLEL across the component
• Should have very high resistance (ideal = ∞ Ω) to draw minimal current
• Converts to show potential difference in Volts

Potentiometer:

A device for measuring potential difference without drawing current (since it is a null method):

• Works on principle of uniform potential gradient along a uniform wire
• Unknown emf compared with standard emf
• More accurate than voltmeter for measuring emf

⚡ NABTEB Quick Reference:

• $I = Q/t$ (current = charge/time)
• $V = IR$ (Ohm’s Law)
• $P = IV = I^2R = V^2/R$ (power)
• $H = I^2Rt$ (Joule’s Law — heating)
• Series: $R_T = R_1 + R_2 + R_3$; $V$ divides; $I$ same
• Parallel: $1/R_T = 1/R_1 + 1/R_2 + 1/R_3$; $I$ divides; $V$ same
• $\varepsilon = V + Ir$ (battery with internal resistance)
• $R = \rho l/A$ (resistivity)
• $R_T = R_0(1 + \alpha \Delta T)$ (temperature dependence)