Chapter 42: Current, Voltage, Resistance, and Ohm’s Law

42.1 From Static Charge to Moving Charge

In Chapters 38–41, electric charge was treated as stationary — charges sitting on surfaces, repelling and attracting, producing electric fields and storing energy in capacitors. Electrostatics is the physics of charges at rest. Current electricity is the physics of charges in continuous, directed motion. The transition from static to current electricity requires a driving mechanism. Isolated charges on a surface eventually reach equilibrium and stop redistributing. For charge to flow continuously, there must be a sustained difference in electric potential (voltage) between two points connected by a conducting path. A battery maintains this potential difference by chemical reactions; a generator maintains it by electromagnetic induction. As long as the potential difference is sustained and the path is complete, charge flows. This continuous flow of charge is what we call an electric current. Understanding it requires introducing a precise definition of current and distinguishing it from related quantities such as drift speed and charge carrier density.

Alessandro Volta invented the first practical battery (the voltaic pile) in 1800, providing physicists with the first sustained source of electric current. Before this, the only available electricity was static charge from friction machines and Leyden jars. Volta's invention sparked (literally and figuratively) two centuries of electrical science and technology. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.2 Electric Current

Electric current (I) is defined as the rate of flow of electric charge through a cross-

section of a conductor. If charge ΔQ passes through a cross-section in time Δt, the

average current is:

I = ΔQ / Δt

Current tells us how many coulombs of charge pass through a wire per second. A current of 1 A means 1 coulomb flows through the cross-section every second. Since

each electron carries charge e = 1.602 × 10 ¹⁹ C, one ampere corresponds to roughly

6.24 × 10¹⁸ electrons passing per second — an enormous number. Device / Situation Typical Current Nerve impulse (biological)

~1 μA (microampere)

LED indicator light ~20 mA Smartphone charging 1–2 A Household light bulb (60 W) ~0.5 A Electric kettle ~10 A Car starter motor 100–200 A Lightning bolt ~30 000 A (peak) Industrial arc furnace ~100 000 A

Table 42.1

Current is a scalar in simple circuit analysis (it has a signed direction along the wire, but no spatial vector direction the way force does). In the International System, the ampere is one of the seven base units, defined since 2019 in terms of the elementary charge e. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.3 Conventional Current vs. Electron Flow

By historical convention — established before the discovery of the electron — electric current is defined as flowing from the positive terminal of a source to the negative terminal through the external circuit. This is conventional current direction. In a metallic conductor, the actual charge carriers are electrons, which are negatively charged. Electrons are attracted toward the positive terminal and repelled from the negative terminal, so they flow in the direction opposite to conventional current. Conventional current flows left to right; electrons flow right to left in the same wire.

is used in all circuit analysis, Ohm’s Law, and engineering practice. Electron flow direction is the physical reality but is rarely used in circuit calculations. Always use conventional current unless explicitly told otherwise. In some conductors, positive charge carriers do flow in the conventional direction. Ionic solutions (electrolytes) carry current by positive ions moving toward the negative electrode and negative ions moving toward the positive electrode. Semiconductor holes (missing electrons) also behave as positive charge carriers moving in the conventional direction. In all cases, the circuit analysis rules are the same. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.4 Voltage as Electric Potential Difference

Voltage (V), more precisely called electric potential difference, is the work done per unit charge in moving a positive test charge between two points in an electric field. It is the electrical analogue of height difference in a gravitational field — it describes the energy landscape that drives charge flow.

V = W / Q

Voltage does not describe how much charge has moved, but rather how much energy is available per coulomb of charge. A 9 V battery delivers 9 joules of energy for every coulomb of charge that flows through it. A 1.5 V cell delivers 1.5 J/C. The voltage tells you the energy "density" per unit charge, not the total energy. In a circuit, voltage is always measured between two points — it is a difference, not an absolute value. We speak of the voltage across a resistor, across a battery, or across a lamp. Saying "the voltage at point A" is only meaningful if a reference point (usually called ground, at 0 V) has been established. Voltage is what drives current through a circuit. Without a potential difference between two connected points, no net charge flow occurs. A wire connected to both terminals of a battery sits in a voltage difference; a wire with both ends at the same potential carries no current.

A 12 V car battery delivers 600 C of charge to the starter motor during starting. Energy delivered = V × Q = 12 × 600 = 7200 J = 7.2 kJ. This energy came from the battery's chemical energy. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law Source Typical Voltage Single electrochemical cell (AA, AAA) 1.5 V Car lead-acid battery (12 cells) 12 V Household mains (UK) 230 V (RMS) Household mains (US) 120 V (RMS) High-voltage transmission lines 100 000 – 750 000 V Static spark (walking on carpet) ~10 000 – 30 000 V Lightning bolt ~300 million V

Table 42.2

Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.5 Resistance

Resistance (R) is the property of a conductor that opposes the flow of electric current. Every material — even a copper wire — has some resistance, though metals have very low resistance. Insulators have extremely high resistance. Resistance converts electrical energy to heat as charge carriers collide with atoms in the conducting material.

R = V / I

Physically, resistance arises from collisions between moving electrons and the fixed lattice of atoms in the conductor. As temperature increases, atoms vibrate more vigorously, increasing the rate of collisions and therefore increasing resistance in most metals. This is why a light bulb filament has different resistance when cold versus when glowing hot. Component Typical Resistance Copper wire (short household wire)

<0.01 Ω

Household light bulb (60 W, hot)

~880 Ω

Typical resistor (electronics)

10 Ω – 10 MΩ

Human body (skin-to-skin, dry)

100 000 – 1 000 000 Ω

Human body (wet skin)

1 000 – 10 000 Ω

Rubber insulation

10¹² – 10¹⁶ Ω

Silicon semiconductor (intrinsic)

~640 Ω·m (resistivity)

Table 42.3

Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.6 Factors Affecting Resistance

The resistance of a uniform conductor depends on four factors: the material it is made

of (characterised by resistivity ρ), its length L, its cross-sectional

area A, and its temperature.

R = ρL / A

Resistivity (ρ) is a material property measuring how strongly the material opposes

current flow. Copper has ρ

1.7 × 10 ⁸ Ω·m; nichrome (used in heating elements) has

ρ

1.1 × 10 ⁶ Ω·m — about 65 times higher. The extremely high resistivity of

insulators like rubber (~10¹³ Ω·m) explains why they are used to prevent current flow.

Length: doubling the length of a wire doubles its resistance (more collisions along a longer path). Cross-sectional area: doubling the area halves the resistance (more parallel paths for charge to flow). This is why high-current cables (e.g., for car batteries or industrial machinery) are thick — large A reduces R, reducing energy lost as heat. Temperature: for metals, resistance increases roughly linearly with temperature above

room temperature: R(T) = R₀[1 + α(T

T₀)], where α is the temperature coefficient of resistance. For semiconductors, resistance typically decreases as temperature rises (more charge carriers are thermally excited). For superconductors below their critical temperature, resistance drops to exactly zero.

A copper wire (resistivity 1.72 × 10 ⁸ Ω·m) is

2.0 m long with diameter 1.0 mm. Cross-sectional area A = π(0.5 × 10 ³)² = 7.85 ×

10 ⁷ m². R = ρL/A = (1.72 × 10 ⁸)(2.0)/(7.85 × 10 ⁷) = 0.044 Ω. Very low, as expected for a short copper wire. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.7 Ohm’s Law

Ohm’s Law states that, for many conducting materials and devices at constant temperature, the current through the conductor is directly proportional to the voltage across it. The constant of proportionality is the resistance.

V = IR

Ohm’s Law is not a universal law of physics in the way Newton’s Laws are. It is an empirical relationship that holds for a class of materials called ohmic conductors: metals, metallic alloys, and many resistors over a wide range of voltages and currents, provided temperature is constant. For ohmic conductors, a graph of V vs. I is a straight line passing through the origin, with slope equal to R. Non-ohmic devices — discussed in section 42.11 — do not obey this linear relationship. Diodes, light-emitting diodes, transistors, thermistors, and light bulbs (when their temperature changes significantly) all have non-linear V–I characteristics. Ohm’s Law still applies instantaneously at any single operating point (giving an effective resistance at that point), but R is not constant.

Ohm’s Law: V = IR. Three variables, three forms. V = IR (find voltage). I =

V/R (find current). R = V/I (find resistance). All three are “Ohm’s Law”; memorise all

three forms.

Georg Simon Ohm published his law in 1827, derived from careful experimental measurements on metal wires. His result was initially dismissed by German scientific authorities, who preferred theoretical derivation to experimental measurement. He was eventually vindicated, and the unit of resistance bears his name. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.8 Using Ohm’s Law to Solve Circuit Problems

Ohm’s Law problems always involve three quantities (V, I, R) with two given and one unknown. The standard method is:

Step 1: Identify: List the given quantities and their units. Identify what is unknown.

Step 2: Choose: Select the correct form of Ohm’s Law (V = IR, I = V/R, or R = V/I).

Step 3: Substitute: Insert values with units. Carry units through the calculation.

Step 4: Check: Does the answer have the right unit? Is the magnitude reasonable?

A 240 Ω resistor is connected to a 12 V battery. Find the

current. I = V/R = 12/240 = 0.050 A = 50 mA.

R = V/I = 6.0/0.25 = 24 Ω.

A current of 3.5 A passes through a 15 Ω resistor. The

voltage across it: V = IR = 3.5 × 15 = 52.5 V.

not try to remember “which formula to use” — start from V = IR every time and

rearrange as needed. This reduces errors and keeps reasoning transparent. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.9 Power in Electric Circuits

Electric power (P) is the rate at which electrical energy is transferred in a circuit — the energy per unit time. It tells us how quickly a device converts electrical energy to other forms (heat, light, mechanical energy, etc.).

P = IV

Using V = IR, we can substitute into P = IV to obtain two additional forms:

P = I²R

P = V²/R

Power is always positive in a resistor — resistors always dissipate energy, never supply it. The dissipated energy heats the resistor. An electric heater intentionally dissipates large power as heat. A microprocessor must dissipate power as waste heat even though its purpose is computation, which is why computers need cooling fans or heat sinks.

A 60 W light bulb operates at 120 V. (a) Current: I = P/V = 60/120 = 0.50 A. (b) Resistance: R = V/I = 120/0.50 = 240 Ω. Check: P = V²/R =

120²/240 = 14400/240 = 60 W. ✓

Device Typical Power Current at 120 V LED bulb (equivalent to 60 W incandescent) 9 W 0.075 A Incandescent 60 W bulb 60 W 0.50 A Desktop computer 150–350 W 1.25–2.9 A Microwave oven 700–1200 W 5.8–10 A Electric kettle 1500–2400 W 12.5–20 A Air conditioner (window unit) 1000–3500 W 8.3–29 A Electric car charger (Level 2) 7200–11 500 W (at 240 V) 30–48 A Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

Table 42.4

42.10 Energy Transfer in Circuits

Electrical energy (E) is power multiplied by time. If a device operates at constant power P for time t, the total electrical energy transferred is:

E = Pt = IVt

The kilowatt-hour is the practical unit of electrical energy used by utility companies. One kWh is the energy consumed by a 1000 W device running for one hour, or equivalently a 100 W device for 10 hours. If electricity costs $0.15 per kWh, a 100 W light bulb running 24 hours costs: energy = 0.1 kW × 24 h = 2.4 kWh; cost = 2.4 ×

$0.15 = $0.36.

An electric kettle (P = 2000 W) heats water for 3

minutes. Energy = 2000 × (3 × 60) = 360 000 J = 360 kJ = 0.10 kWh. At $0.15/kWh,

cost = $0.015 = 1.5 cents per boil.

In any circuit, energy is conserved. Energy delivered by the source (battery, generator) equals energy dissipated in all resistors plus energy stored in capacitors and inductors. For a simple resistive circuit with no storage elements, all source energy is converted to heat in the resistors. This is the basis for circuit efficiency analysis in later chapters. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.11 Real Resistors and Non-Ohmic Devices

An ideal ohmic resistor has constant resistance regardless of the voltage or current applied. Real resistors approximate this behaviour over a specified operating range. Exceed that range (too much power, too high voltage) and resistance changes, the resistor overheats, or it fails. Non-ohmic devices have resistance that depends on the applied voltage or current. Their V–I graphs are not straight lines through the origin. Key examples include: • Diode: allows current to flow easily in one direction (forward bias, very low resistance) but blocks it in the other direction (reverse bias, very high resistance). Used in rectifiers that convert AC to DC. • Light-emitting diode (LED): like a diode but emits light when forward-biased. Requires a current-limiting resistor in series because resistance drops sharply once conduction begins. • Thermistor: resistance decreases sharply as temperature increases (NTC type) or increases with temperature (PTC type). Used in temperature sensors and self- resetting fuses. • Light-dependent resistor (LDR / photoresistor): resistance decreases as light intensity increases. Used in automatic street lights and camera exposure systems. • Filament lamp: resistance increases significantly as temperature rises. A cold filament has much lower resistance than when glowing — so the initial surge current is much higher than the steady-state current.

The formula R = V/I always gives the resistance at a specific

operating point, even for non-ohmic devices. But for non-ohmic devices, this R is not constant — it changes with V and I. Never apply Ohm’s Law to a non-ohmic device over a range of voltages and assume R is constant. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

42.12 Common Mistakes in Current and Voltage Problems

• Confusing current and voltage. Current (I) is the rate of charge flow (A). Voltage (V) is the energy per unit charge (V). They are fundamentally different quantities. A high-voltage source can deliver low current, and a low-voltage source can deliver high current. Describe each explicitly. •

Using Ohm’s Law with non-ohmic devices over a range. Ohm’s Law V = IR holds

for ohmic devices at constant temperature. Applying it to a diode or LED over a

range of voltages gives wrong results. At a single operating point, R = V/I is still

the instantaneous resistance, but it is not constant. •

Forgetting to convert units. Milliamperes to amperes (1 mA = 10 ³ A), kilohms to

ohms (1 kΩ = 1000 Ω), milliwatts to watts (1 mW = 10 ³ W). Mixing units gives

wrong numerical answers even with the correct formula. •

Mixing up P = IV, P = I²R, and P = V²/R. All three are correct, but each requires

specific known quantities. If you know V and R, use P = V²/R directly; do not

first compute I from Ohm’s Law unnecessarily. Choose the form that uses the given quantities. • Treating voltage as a property of a single point without a reference. Voltage is always a difference between two points. "The voltage of the resistor" means the voltage across both terminals. If the problem says "voltage at point A" it implies a defined ground (0 V) reference. • Confusing energy and power. Power (W) is rate; energy (J or kWh) is total. A 100

W bulb consuming energy for 10 hours uses 1 kWh = 3.6 MJ. Do not state the

energy as "100 W" or the power as "1 kWh."

Always check units at the end of every calculation. V/A = Ω. A×Ω

= V. V/Ω = A. W = A×V = A²×Ω = V²/Ω. If your units do not work out, the formula is being applied incorrectly. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

Chapter Summary

Electric current, voltage, and resistance are the three fundamental quantities of circuit analysis. Ohm’s Law links them for ohmic devices; power and energy formulas extend the analysis to energy transfer. •

Current I = ΔQ/Δt; unit ampere (A = C/s). Conventional current flows from + to

through external circuit; electrons flow in the opposite direction. •

Voltage V = W/Q; unit volt (V = J/C). It is a potential difference — always

measured between two points. Voltage drives current through a circuit. •

Resistance R = V/I; unit ohm (Ω = V/A). Depends on material resistivity, length,

cross-sectional area, and temperature: R = ρL/A.

•

Ohm’s Law: V = IR (or I = V/R or R = V/I). Valid for ohmic conductors at

constant temperature. •

Power: P = IV = I²R = V²/R; unit watt (W = J/s).

•

Energy: E = Pt; practical unit kilowatt-hour (1 kWh = 3.6 MJ).

• Non-ohmic devices (diodes, LEDs, thermistors, filament lamps) have non- constant resistance; Ohm’s Law does not apply over a range of operating conditions. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

Key Terms

electric current (I)

Rate of charge flow through a cross-section:

I = ΔQ/Δt; unit

ampere (A).

conventional current

Current direction defined as positive-to-negative through the

external circuit; opposite to the direction of electron flow.

voltage (V)

Electric potential difference; energy per unit charge: V = W/Q; unit volt (V

= J/C). Always measured between two points.

resistance (R)

Opposition to current flow: R = V/I; unit ohm (Ω). Depends on material,

geometry, and temperature.

resistivity (ρ)

Material property quantifying intrinsic opposition to current: R = ρL/A;

unit Ω·m.

Ohm's

Law

V = IR; for

ohmic conductors at constant temperature, current is proportional to voltage.

ohmic conductor

Material or device for which resistance is constant over a range of

voltages and currents; V–I graph is a straight line through the origin.

non-ohmic device

Device whose resistance varies with applied voltage or current;

diodes, LEDs, thermistors, and filament lamps are examples.

electric power (P)

Rate of energy transfer: P = IV = I²R = V²/R; unit watt (W = J/s).

electrical energy (E)

Energy transferred: E = Pt = IVt; practical unit kilowatt-hour (kWh);

1 kWh = 3.6 MJ.

drift

velocity

Average velocity of charge carriers along a conductor; typically millimetres per second in metals despite the near-instant electrical signal propagation.

Physics

Foundations of Physical Science

Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

Concept Review Questions

Answer in complete sentences. No calculations required unless stated.

1. A current of 1 A flows through a wire. How many electrons pass through a cross

-section of the wire each second? (Use e = 1.6 × 10 ¹⁹ C.)

2. Explain in your own words the difference between conventional current direction

and the direction of electron flow. Why do we use conventional current in circuit analysis even though electrons flow the opposite way?

3. A wire and a resistor are connected to the same battery. The wire has negligible

resistance. Explain why the voltage across the wire is nearly zero while the voltage across the resistor is nearly the full battery voltage.

4. Two resistors are made of the same material. Resistor A is twice as long as

resistor B but has the same cross-sectional area. How do their resistances

compare? Explain using R = ρL/A.

5. A light bulb filament has a much higher resistance when glowing than when

cold. Does it obey Ohm’s Law? Explain.

6. Why do high-current cables (such as those in car engines) need to be thick? Use

the resistance formula to explain.

7. A 100 W light bulb and a 40 W light bulb are both rated at 120 V. Which has

higher resistance? Which carries more current? Explain.

8. Explain the difference between power and energy. A student says "the light bulb

used 60 watts of energy in one hour." What is wrong with this statement?

Current Problems

9. A battery delivers 450 C of charge in 3.0 minutes. Find the average current.
10. A current of 2.5 A flows through a wire for 4.0 s. How many coulombs of charge

pass through the wire? How many electrons is this?

11. A charge of 0.18 C flows through a resistor in 90 ms. Find the current in

milliamperes.

12. A nerve impulse carries a current of 1.5 μA for 2.0 ms. Find the charge

transferred.

13. An LED requires a forward current of 20 mA. If it is operated for 8 hours per day

for 30 days, find the total charge transferred through the LED. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

Ohm’s Law Problems

14. A 470 Ω resistor is connected to a 9.0 V battery. Find the current through it.
15. A current of 0.15 A flows through a resistor when 6.0 V is applied. Find the

resistance.

16. A resistor of 1200 Ω carries a current of 22 mA. Find the voltage across it.
17. A copper wire has resistivity 1.72 × 10 ⁸ Ω·m, length 5.0 m, and cross-sectional

area 1.0 mm². Find its resistance. If 12 V is applied across the wire, find the

current.

18. The resistance of a heating element is 28 Ω at room temperature and 42 Ω at

operating temperature. If connected to a 240 V supply, find the steady-state current. What is the initial cold-start current?

Electric Power Problems

19. A toaster operates at 120 V and draws 8.0 A. Find its power rating and

resistance.

20. A 1500 W hair dryer is connected to a 120 V outlet. Find the current it draws

and its resistance.

21. A 150 Ω resistor carries a current of 0.40 A. Find the power dissipated.
22. A 240 V, 2.0 kW electric heater operates for 3.5 hours. Find (a) the current drawn,

(b) the resistance of the heating element, and (c) the energy consumed in joules and in kWh.

23. Two resistors are connected to the same 12 V battery. Resistor A has R = 60 Ω;

Resistor B has R = 180 Ω. Find the power dissipated in each. Which dissipates

more power? Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

Applied Circuit Analysis

24. A wire of resistivity 2.8 × 10 ⁸ Ω·m, length 80 m, and diameter 2.0 mm is used

to carry current from a 24 V source. Find (a) the resistance, (b) the current, and (c) the power dissipated in the wire. Is this power loss acceptable for a lighting circuit? Explain.

25. A car headlamp is rated 55 W at 12 V. (a) Find its resistance at operating

temperature. (b) If the battery voltage drops to 10.5 V during starting, find the new current and power. By what percentage does brightness (proportional to power) decrease?

26. A student measures the following V–I data for a device: (1 V, 0.01 A), (2 V, 0.02

A), (3 V, 0.03 A), (5 V, 0.10 A), (7 V, 0.30 A). (a) Is the device ohmic? Justify using the data. (b) Find the resistance at the 3 V operating point and at the 7 V operating point.

27. A household circuit is protected by a 15 A fuse (it opens the circuit if current

exceeds 15 A). The circuit operates at 120 V. (a) What is the maximum total power that can be safely drawn? (b) If a 1200 W microwave, an 800 W toaster, and a 100 W lamp are all on the same circuit, will the fuse blow? Show your calculation. Physics: Foundations of Physical Science Chapter 42 — Current, Voltage, Resistance, and Ohm’s Law

Chapter 42 Checkpoint

Before moving on to Chapter 43, confirm you can do each of the following. I can… See Section Explain how static charge differs from current electricity 42.1

Define current and calculate I = ΔQ/Δt

42.2 Distinguish conventional current from electron flow 42.3

Define voltage as potential difference and calculate V = W/Q

42.4 Define resistance and describe the factors that determine it 42.5–42.6

State Ohm’s Law (V = IR) and use all three rearrangements

42.7–42.8

Calculate electric power using P = IV, P = I²R, P = V²/R

42.9

Calculate electrical energy (E = Pt) and convert between J and

kWh 42.10 Distinguish ohmic from non-ohmic devices and give examples of each 42.11 Identify and avoid common mistakes in current and voltage problems 42.12