DC Circuits

SYLLABUS

Electric Currents; Drift velocity and its relation with current, Ohm’s law; Electrical Resistance; Resistivity; Conductivity, Current-voltage relations; Ohmic and Non-Ohmic resistance, Resistances in series and parallel, Potential divider, Electromotive force of a source, internal resistance, Work and power in electrical circuits

What is This Chapter About?

Switch on a light bulb and it glows. Connect a battery to a motor and it spins. Plug a phone charger into a socket and the battery fills. All of these involve electric current flowing through circuits — carefully arranged paths of conductors, resistors, batteries, and other components that direct and control the flow of charge to do useful work.

Chapter 23 of NEB Grade 11 Physics is the most comprehensive chapter in the electricity section, carrying 10 teaching hours — the highest of any chapter in Grade 11. It develops the complete quantitative theory of direct current circuits: how current flows, what opposes it, how components combine, how circuits are powered, and how energy is transferred. These concepts are the foundation of all practical electrical engineering and the theoretical basis for understanding every electronic device ever built.

Key Concepts You Will Learn

Electric Current and Drift Velocity

When a potential difference is applied across a conductor, free electrons experience an electric force and begin moving — but not freely. They constantly collide with positive ions in the crystal lattice, giving up energy as heat. Between collisions, they accelerate; immediately after each collision, they start again from near rest. The result is a slow net drift of electrons in one direction — the drift velocity ([latex]v_d[/latex]), typically of the order of millimetres per second in a wire carrying normal current.

Electric current is defined as the rate of flow of positive charge: [latex]I = \frac{dQ}{dt}[/latex]. The direction of conventional current is taken as the direction positive charges would flow — opposite to the actual electron drift direction. The SI unit of current is the ampere (A).

The relationship between current and drift velocity is derived by considering how many electrons pass a cross-section per second:

[latex]I = nAvq[/latex]

where n is the number density of free charge carriers (carriers per unit volume), A is the cross-sectional area of the conductor, v is the drift velocity, and q is the charge of each carrier. This fundamental result connects the macroscopic current to microscopic charge carrier motion.

Ohm’s Law and Electrical Resistance

Ohm’s Law states that for many conductors at constant temperature, the current through the conductor is directly proportional to the potential difference across it:

[latex]V = IR[/latex]

where R is the electrical resistance of the conductor — its opposition to current flow. The SI unit of resistance is the ohm (Ω) = volt per ampere.

Resistivity ([latex]\rho[/latex]) is a material property that measures how strongly a material resists current flow, independent of the conductor’s dimensions:

[latex]R = \frac{\rho l}{A}[/latex]

where l is the length and A is the cross-sectional area. Long, thin wires have high resistance; short, thick wires have low resistance. The inverse of resistivity is conductivity ([latex]\sigma = \frac{1}{\rho}[/latex]).

Current-Voltage Characteristics — Ohmic and Non-Ohmic

An ohmic conductor (like a metal wire at constant temperature) obeys Ohm’s Law — its I-V graph is a straight line through the origin. A non-ohmic component does not obey Ohm’s Law — its I-V graph is non-linear. Important non-ohmic components include semiconductor diodes (exponential I-V curve), filament lamps (resistance increases as temperature rises, so I-V curves upward), and thermistors. Understanding and sketching I-V characteristics for different components is an important skill.

Resistances in Series and Parallel

For resistors in series (same current flows through each), the equivalent resistance is the sum:

[latex]R_s = R_1 + R_2 + R_3 + \ldots[/latex]

The voltage divides between them in proportion to their resistances. For resistors in parallel (same voltage across each), the reciprocal of equivalent resistance equals the sum of reciprocals:

[latex]\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots[/latex]

The current divides between parallel branches in inverse proportion to their resistances. These combination rules — derived from conservation of charge and energy — allow any resistor network to be simplified.

Potential Divider

A potential divider is a circuit consisting of two (or more) resistors connected in series across a supply voltage, with the output voltage taken across one of the resistors:

[latex]V_{out} = V_{in} \times \frac{R_2}{R_1 + R_2}[/latex]

The potential divider principle is used extensively in sensor circuits. A thermistor (resistance decreases with temperature) in a potential divider gives a voltage that varies with temperature — used in electronic thermometers. An LDR (light-dependent resistor, resistance decreases with light intensity) in a potential divider gives a voltage that varies with illumination — used in automatic lighting systems. A strain gauge in a potential divider gives a voltage proportional to strain — used in weighing scales and structural monitoring.

Electromotive Force and Internal Resistance

A battery (or any source of electrical energy) has two key properties. Its electromotive force (EMF) ([latex]\mathcal{E}[/latex]) is the energy supplied per unit charge driven around a complete circuit — it represents the energy input from the chemical reaction (or other energy source) per coulomb of charge. The unit of EMF is the volt, like potential difference, but the two concepts are distinct: EMF is an energy source, while potential difference is an energy drop.

Every real battery also has internal resistance (r) — resistance within the battery itself due to the electrolyte and electrode materials. When current I flows, the terminal potential difference (the voltage available to the external circuit) is:

[latex]V_{terminal} = \mathcal{E} – Ir[/latex]

The term Ir represents the voltage drop across the internal resistance — energy lost as heat inside the battery. This is why a battery’s terminal voltage drops when current increases (heavy load) and recovers when the circuit is open. Measuring terminal voltage under load and without load allows both EMF and internal resistance to be determined.

Work and Power in Electrical Circuits

The electrical power delivered to any component is:

[latex]P = IV = I^2R = \frac{V^2}{R}[/latex]

These three equivalent forms are all derivable from [latex]P = IV[/latex] combined with Ohm’s Law. The total electrical energy dissipated in time t is:

[latex]W = Pt = IVt = I^2Rt[/latex]

For a resistor, this energy appears as heat — Joule heating. The rate of heat production ([latex]P = I^2R[/latex]) is the basis of electric heaters, toasters, incandescent lamps, and fuse wires. For a motor, some of the electrical energy is converted to mechanical work rather than heat — the motor is more efficient than a resistor for this reason.

Sensors in Potential Divider Circuits

The potential divider circuit combined with sensors (thermistors, LDRs, strain gauges) creates transducers — devices that convert non-electrical quantities (temperature, light, pressure) into electrical signals that can be processed by electronic circuits. Understanding how each sensor works and how to calculate output voltages for given sensor resistances is an important applied skill.

Why This Chapter Matters for NEB Exams

DC Circuits is the highest-weightage chapter in Grade 11 Physics and is examined comprehensively in every NEB board paper. Numerical problems on Ohm’s Law, resistivity, series and parallel combinations, and power calculations appear in every examination. The derivation of the current-drift velocity relation

[latex]I = nAvq[/latex]

is a classic long-answer question. EMF and internal resistance — finding them from terminal voltage measurements and solving circuit problems with internal resistance — appear regularly. Potential divider calculations and sensor applications are standard short and medium-answer questions. I-V characteristics of ohmic and non-ohmic components are common drawing and explanation questions.

Important Topics to Focus On

• Electric current — definition and direction (conventional vs electron flow)
• Drift velocity and its relation to current: [latex]I = nAvq[/latex]
• Ohm’s Law: [latex]V = IR[/latex] — statement and conditions
• Resistance, resistivity, and conductivity:

[latex]R = \frac{\rho l}{A}[/latex]

I-V characteristics — ohmic (metallic conductor) and non-ohmic (diode, filament lamp) Resistances in series: [latex]R_s = \sum R_i[/latex] Resistances in parallel:

[latex]\frac{1}{R_p} = \sum \frac{1}{R_i}[/latex]

Potential divider principle and formula Sensors in potential divider circuits — thermistor, LDR, strain gauge EMF vs potential difference — conceptual distinction Internal resistance — terminal voltage:

[latex]V = \mathcal{E} – Ir[/latex]

Power in circuits:

[latex]P = IV = I^2R = \frac{V^2}{R}[/latex]

Joule heating and its applications Numerical problems on all of the above

Connection to Other Chapters

DC Circuits builds on Chapter 19 (Electric Charges) — charge carriers and their properties — Chapter 20 (Electric Field) — the field drives drift of charges — and Chapter 21 (Potential) — potential difference drives current. Chapter 22 (Capacitor) adds energy storage to the circuit picture. In Grade 12, Chapter 14 (Electrical Circuits) extends DC circuit analysis to complex networks using Kirchhoff’s Laws, Wheatstone bridge, and potentiometer. Chapter 19 (Alternating Currents) replaces DC with sinusoidal AC and introduces reactance and impedance. The power formula

[latex]P = I^2R[/latex]

connects to Chapter 18 (Electromagnetic Induction) through transformer losses, and to the concept of power factor in AC circuits. The potential divider and sensor circuits developed here are the foundation of all electronic sensing and measurement systems used in physics experiments throughout both Grade 11 and Grade 12.