Unit 4

Electricity and Magnetism

Class 12 Physics

Chapter number not exposed on the legacy source page

D.C. Circuit

On mobile, swipe inside the PDF to read all pages and pinch to zoom.

Open Original Nepal eNotes Page

Contents / Quick Navigation

Introduction
Current & Drift
Ohm’s Law
Resistance
Series & Parallel
Potential Divider
Meters
EMF & Internal Resistance
Work, Power & Joule Heating
Exam Questions

Introduction

A direct-current (D.C.) circuit is an electrical circuit in which charge has a steady preferred direction of flow. In the NEB/CDC syllabus, the D.C. circuit topic covers electric current, drift velocity, Ohm’s law, resistance and resistivity, conductivity, combinations of resistors, potential divider, measuring instruments, electromotive force and internal resistance, electrical work and power, and Joule heating.

Definition: Electric current is the rate of flow of electric charge through a cross-section of a conductor. I = dQ/dt (or I = Q/t for steady current) SI unit: ampere (A) = coulomb per second (C s⁻¹).

Electric Current, Current Density and Drift Velocity

In a metallic conductor, free electrons have rapid random thermal motion. When an electric field is applied, this random motion acquires a very small average velocity called drift velocity, directed opposite to the conventional current.

Key relations I = n e A v_d J = I/A = n e v_d where n = number density of free electrons (m⁻³), e = magnitude of electron charge (C), A = cross-sectional area (m²), v_d = drift speed (m s⁻¹) and J = current density (A m⁻²).

Diagram 1: Drift of electrons

Conventional current is opposite to electron drift in a metal.

Worked Example 1. A copper wire carries 2.0 A. If n = 8.5 × 10²⁸ m⁻³ and its cross-sectional area is 1.0 mm², find the drift speed.

Solution: A = 1.0 × 10⁻⁶ m². Using v_d = I/(neA):

v_d = 2/[8.5×10²⁸ × 1.6×10⁻¹⁹ × 10⁻⁶] ≈ 1.47×10⁻⁴ m s⁻¹

Unit check: A/(m⁻³·C·m²) = (C s⁻¹)/(C m⁻¹) = m s⁻¹.

Ohm’s Law and I–V Characteristics

Ohm’s law: At constant physical conditions, especially temperature, the current through a metallic conductor is directly proportional to the potential difference across it. V ∝ I ⇒ V = IR

An ohmic conductor has a straight-line I–V graph through the origin at constant temperature. A diode, filament lamp and many semiconductor devices are non-ohmic because their resistance is not constant over the operating range.

Diagram 2: Ohmic and non-ohmic behaviour

A straight-line I–V relation indicates constant resistance.

Common mistake: Ohm’s law is not a universal law for every device. It applies only when the physical state of the conductor remains effectively constant.

Resistance, Resistivity, Conductivity and Temperature

Resistance of a uniform conductor R = ρL/A where ρ is resistivity (Ω m), L is length (m), and A is cross-sectional area (m²). σ = 1/ρ Conductivity σ has SI unit S m⁻¹ (siemens per metre).

Quantity	Symbol	Meaning	SI unit
Resistance	R	Opposition offered by a particular conductor	Ω
Resistivity	ρ	Material property independent of specimen dimensions	Ω m
Conductance	G	Reciprocal of resistance, G = 1/R	S
Conductivity	σ	Reciprocal of resistivity, σ = 1/ρ	S m⁻¹

Temperature dependence

For many metals over a moderate temperature range:

R = R₀[1 + α(T − T₀)]

where α is the temperature coefficient of resistance (K⁻¹). Semiconductors usually show the opposite trend: their resistance decreases markedly as temperature rises.

Perfect conductor vs superconductor: A perfect conductor is an idealized material with zero resistivity. A superconductor is a real material that, below a critical temperature and under suitable conditions, exhibits zero dc electrical resistance along with other characteristic electromagnetic behaviour.

Resistors in Series and Parallel

Series combination

The same current passes through every resistor. Potential differences add.

R_s = R₁ + R₂ + …

Parallel combination

The potential difference across each branch is the same. Branch currents add.

1/R_p = 1/R₁ + 1/R₂ + …

Diagram 3: Series and parallel networks

Equivalent-resistance relations for two common resistor networks.

Worked Example 2. Find the equivalent resistance of 6 Ω and 3 Ω connected in parallel. 1/R = 1/6 + 1/3 = 3/6 = 1/2 ⇒ R = 2 Ω

Potential Divider

Two or more series resistors can divide a supply voltage. For two resistors R₁ and R₂, with output taken across R₂:

V_out = V × R₂/(R₁ + R₂)

Diagram 4: Potential-divider circuit

Output voltage across R₂ in an unloaded divider.

Exam important: State clearly that the simple divider formula assumes the output is not significantly loaded. A low-resistance load changes the effective resistance of the lower branch.

Galvanometer, Ammeter, Voltmeter and Ohmmeter

A galvanometer is a sensitive current detector. It can be converted into an ammeter by connecting a low resistance shunt in parallel, and into a voltmeter by connecting a large resistance in series.

Galvanometer to ammeter: S = I_gG/(I − I_g) Galvanometer to voltmeter: R = V/I_g − G where G is galvanometer resistance, I_g is full-scale galvanometer current, I is desired ammeter range, and V is desired voltmeter range.

Diagram 5: Converting a galvanometer

A shunt makes an ammeter; a high series resistance makes a voltmeter.

An ohmmeter measures resistance using an internal source and meter movement/electronic measuring circuit. Resistance should normally be measured with the component isolated from external power.

Electromotive Force and Internal Resistance

EMF (E): Energy supplied by a source per unit charge in driving charge around the complete circuit. SI unit: volt (V = J C⁻¹).

A practical cell has internal resistance r. When it supplies current I through external resistance R:

I = E/(R + r) V = IR = E − Ir

Diagram 6: Real cell and internal resistance

Terminal voltage falls below the emf when the cell supplies current.

Worked Example 3. A cell of emf 1.5 V and internal resistance 0.20 Ω is connected to 2.8 Ω. I = 1.5/(2.8+0.2) = 0.50 A V = IR = 0.50×2.8 = 1.40 V

Check: E − Ir = 1.50 − 0.50×0.20 = 1.40 V.

Electrical Work, Power and Joule’s Law

W = VIt = I²Rt = V²t/R P = VI = I²R = V²/R

Electrical work/energy is measured in joule (J); power in watt (W). Commercial electrical energy is commonly measured in kilowatt-hour:

1 kWh = 3.6 × 10⁶ J

Joule’s law of heating: At constant resistance, heat produced is directly proportional to the square of current, resistance and time: H = I²Rt

Worked Example 4. A 12 Ω heater carries 3 A for 5 min. Find heat produced.

t = 300 s.

H = 3²×12×300 = 32,400 J = 32.4 kJ

Important Observations and Common Mistakes

Conventional current is defined in the direction positive charge would move; electron drift is opposite.
Do not confuse emf with terminal potential difference; they are equal only when no current is drawn (or internal resistance is negligible).
In a series circuit current is common; in parallel branches voltage is common.
Resistance depends on geometry and material; resistivity is fundamentally a material property at a given state/temperature.
For power formulas, choose the form that matches known quantities and check watt = joule per second.
Do not apply the simple potential-divider ratio without considering loading when a load is connected.

Important Exam Questions

Short-answer

Define drift velocity and derive the relation I = neAv_d.
Distinguish resistance and resistivity with SI units.
What is an ohmic conductor? Give one non-ohmic example.
Define emf and internal resistance of a cell.
Why is an ammeter connected in series and a voltmeter in parallel?

Long-answer / derivation

Derive the equivalent resistance for series and parallel combinations.
Explain the potential-divider principle and derive the output-voltage relation.
Derive the formula for converting a galvanometer into an ammeter and into a voltmeter.
Derive the current and terminal-voltage relations for a cell of emf E and internal resistance r.
State and explain Joule’s law of heating and obtain the common power relations.

Numerical practice

A 2.0 m wire of resistivity 1.7×10⁻⁸ Ω m has area 0.50 mm². Find its resistance.
A 2 V cell with r = 0.5 Ω supplies a 3.5 Ω load. Find current, terminal voltage and power in the load.
A galvanometer of 100 Ω gives full-scale deflection at 1 mA. Find the shunt for a 1 A ammeter and series resistance for a 10 V voltmeter.

Diagram questions

Draw I–V graphs for an ohmic conductor and a non-ohmic device.
Draw a labelled potential-divider circuit.
Draw galvanometer-to-ammeter and galvanometer-to-voltmeter connections.

One-Minute Revision

I = Q/t and J = I/A.
I = neAv_d.
Ohm’s law: V = IR under constant physical conditions.
R = ρL/A; σ = 1/ρ.
Series: resistances add.
Parallel: reciprocals add.
Potential divider: Vout = V R₂/(R₁+R₂).
Real cell: I = E/(R+r), V = E−Ir.
P = VI = I²R = V²/R.
Joule heat: H = I²Rt.
Ammeter: low resistance, series connection.
Voltmeter: high resistance, parallel connection.

Diagram Practice

Electron drift and conventional current
I–V characteristics
Series/parallel resistors
Potential divider
Cell with internal resistance
Galvanometer conversions

Source handling: The original Nepal eNotes PDF is embedded above using the verified Google Drive file. The typed section follows the verified NEB/CDC syllabus and is designed as a searchable, responsive study companion. Where the PDF viewer does not expose handwritten page text, the typed section is a syllabus-aligned reconstruction and is not claimed to be a word-for-word transcription.

Reference pages: Nepal eNotes source page; Curriculum Development Centre Physics curriculum/resource listings; CDC Grade 12 Physics textbook.