understand that an electric current is a flow of charge carriers

Electric Current and Direct‑Current (D.C.) Circuits

Learning Objectives (Cambridge AS & A‑Level – Physics 9702)

• Define electric current as a flow of charge carriers and express it mathematically.
• Identify the charge carriers in metals, semiconductors, electrolytes and plasmas.
• Distinguish conventional current from electron flow.
• Explain potential difference, electrical power and the three power formulae.
• State Ohm’s law, define resistance and resistivity, and describe their temperature dependence.
• Interpret I‑V characteristics of resistors, filament lamps, diodes, light‑dependent resistors (LDRs) and thermistors.
• Analyse simple D.C. circuits using series/parallel rules, Kirchhoff’s laws and potential‑divider formulae.
• Apply practical techniques for measuring I, V and R and evaluate experimental uncertainties.

1. What is Electric Current?

Electric current (I) is the rate at which electric charge passes a given point in a circuit.

$$I=\frac{\Delta Q}{\Delta t}$$

• ΔQ – charge transferred (coulomb, C)
• Δt – time interval (second, s)
• 1 A = 1 C s⁻¹

2. Charge Carriers in Different Media

Medium	Charge Carrier(s)	Typical Mobility (m² V⁻¹ s⁻¹)
Metals (e.g., Cu)	Free electrons	≈ 4.5 × 10⁻³
n‑type semiconductor	Electrons	≈ 0.1 – 0.2
p‑type semiconductor	Holes	≈ 0.05 – 0.1
Electrolyte (e.g., NaCl solution)	Positive & negative ions	≈ 10⁻⁴ (ions)
Plasma	Ions & free electrons	Varies widely

Conventional Current vs. Electron Flow

• Conventional current: direction a positive charge would move (from higher to lower potential).
• Electron flow: actual motion of electrons, opposite to conventional current in metals.
• Both conventions give the same magnitude of current; the conventional direction is retained for circuit analysis.

3. Potential Difference (Voltage) and Electrical Power

• Potential difference (ΔV): work done per unit charge to move a charge between two points $$\Delta V=\frac{W}{Q}$$
  Unit: volt (V) = joule per coulomb (J C⁻¹).
• Electrical power (P): rate at which electrical energy is transferred $$P = VI = I^{2}R = \frac{V^{2}}{R}$$
  Unit: watt (W) = joule per second (J s⁻¹).

Worked example – A 10 Ω resistor carries 0.25 A. Find the voltage across it and the power dissipated.

$$V = IR = (0.25\;\text{A})(10\;\Omega)=2.5\;\text{V}$$ $$P = VI = (2.5\;\text{V})(0.25\;\text{A})=0.625\;\text{W}$$

4. Resistance, Resistivity and Temperature Effects

• Resistance (R): opposition to the flow of charge; defined by Ohm’s law $$V = IR$$
• Resistivity (ρ): intrinsic property of a material (Ω m). For a uniform conductor $$R = \rho\,\frac{L}{A}$$ where L is length and A is cross‑sectional area.
• Temperature dependence (metals): $$\rho(T)=\rho_{0}\,[1+\alpha\,(T-T_{0})]$$ α ≈ 3.9 × 10⁻³ K⁻¹ for copper.

I‑V Characteristics (syllabus requirement 9.3)

Device	Shape of I‑V curve	Key features
Metallic resistor (Ohmic)	Straight line through the origin	Slope = 1/R, constant resistance.
Filament lamp (non‑Ohmic)	Curve that steepens with increasing V	Resistance rises as temperature rises.
Silicon diode	Near‑zero current until V≈0.6 V (forward), then exponential rise; very small reverse current.	Threshold (cut‑in) voltage ≈ 0.6 V.
Light‑dependent resistor (LDR)	Resistance decreases sharply with increasing illumination; I‑V remains linear for a given light level.	Photoconductivity – R ∝ 1/(light intensity).
Thermistor (NTC)	Resistance falls exponentially with temperature; I‑V stays linear for a fixed temperature.	R = R₀ e^{-βT} (β is material constant).

5. Direct‑Current (D.C.) Circuits

5.1 Series and Parallel Combinations

Configuration	Current (I)	Voltage (V)	Equivalent Resistance (Req)
Series	Same through all components	Sum of individual drops	Req = ΣRi
Parallel	Same across each branch	Sum of branch currents	1/Req = Σ1/Ri

5.2 Kirchhoff’s Laws

1. Kirchhoff’s Current Law (KCL): The algebraic sum of currents entering a junction equals the sum leaving. $$\sum I_{\text{in}} = \sum I_{\text{out}}$$
2. Kirchhoff’s Voltage Law (KVL): The algebraic sum of potential differences round any closed loop is zero. $$\sum V = 0$$

5.3 Potential Divider

Two series resistors R₁ and R₂ produce a fraction of the supply voltage:

$$V_{\text{out}} = V_{\text{supply}}\;\frac{R_{2}}{R_{1}+R_{2}}$$

Used for biasing semiconductor devices and for reference voltages.

5.4 Example Circuit Analysis

Find the current through each resistor in the series circuit shown (12 V battery, R₁=10 Ω, R₂=20 Ω, R₃=30 Ω).

• Total resistance: R_eq = 10 + 20 + 30 = 60 Ω.
• Total current: I = V/R = 12 V / 60 Ω = 0.20 A.
• Voltage across each resistor: V₁ = IR₁ = 2 V, V₂ = 4 V, V₃ = 6 V.

6. Practical Skills (Paper 3 & Paper 5)

7. Summary

• Electric current is the flow of charge carriers; its magnitude is charge per unit time (A = C s⁻¹).
• Charge carriers differ by material: electrons in metals, electrons/holes in semiconductors, ions in electrolytes, both ions and electrons in plasmas.
• Conventional current direction is defined as the direction a positive charge would move; electron flow is opposite in metals.
• Potential difference (voltage) is work per unit charge; electrical power may be expressed as VI, I²R or V²/R.
• Resistance follows Ohm’s law (V = IR); resistivity links resistance to geometry and material (R = ρL/A) and varies with temperature.
• I‑V characteristics: linear for Ohmic resistors, non‑linear for filament lamps, diodes, LDRs and thermistors (see table).
• D.C. circuit analysis uses series/parallel rules, Kirchhoff’s laws and the potential‑divider formula.
• Practical measurements require correct instrument connections, appropriate range selection and careful uncertainty evaluation.

8. Practice Questions

1. Calculate the current if $1.2\times10^{-2}\,\text{C}$ of charge passes a point in $3.0\,\text{s}$.
2. A copper wire (density $8.96\times10^{3}\,\text{kg m}^{-3}$, atomic mass $63.5\,\text{g mol}^{-1}$) has a cross‑sectional area of $1.0\times10^{-6}\,\text{m}^{2}$ and an electron drift speed of $2.2\times10^{-4}\,\text{m s}^{-1}$. Estimate the current flowing through the wire.
3. Explain why the direction of conventional current is opposite to the direction of electron flow in a metal circuit.
4. Given a 12 V battery connected to a series circuit of three resistors (10 Ω, 20 Ω, 30 Ω), find the current through each resistor and the voltage across each.
5. Sketch the I‑V characteristic of a silicon diode and label the forward‑bias threshold.
6. Draw and analyse a circuit containing a 9 V battery, two loops and three resistors (R₁=5 Ω, R₂=10 Ω, R₃=15 Ω) using Kirchhoff’s laws. State the current in each branch.
7. Design a simple experiment to verify the temperature dependence of a metal’s resistance. State the variables, method of measurement and the form of the graph you would plot.
8. For an LDR, the resistance varies from $10 kΩ$ (dark) to $1 kΩ$ (bright). If a 5 V supply is applied across the LDR in series with a 1 kΩ resistor, calculate the voltage across the LDR in both lighting conditions.
9. A thermistor (NTC) obeys $R = R_{0}e^{-\beta T}$ with $R_{0}=5 kΩ$ at $0^{\circ}\text{C}$ and $\beta = 0.02\;\text{K}^{-1}$. Find its resistance at $25^{\circ}\text{C}$.