sketch the I–V characteristics of a metallic conductor at constant temperature, a semiconductor diode and a filament lamp

Resistance, Resistivity & I‑V Characteristics (Cambridge IGCSE / A‑Level)

Learning objectives

• Sketch the I–V characteristics of:

  1. a metallic conductor kept at constant temperature,
  2. a semiconductor diode,
  3. a filament lamp.

• Explain how the shape of each curve is linked to the material’s resistivity and its temperature‑coefficient.
• Use the relevant power formulas and Kirchhoff’s voltage law (KVL) when the devices are placed in simple circuits.
• Identify the correct circuit symbols, e.m.f., internal resistance and the principle of a potential divider (syllabus 10.1).

1. Fundamental definitions (syllabus 9.1 & 9.2)

• Electric current (I) – the rate of flow of charge:

  \$I=\frac{Q}{t}\qquad\text{(A)}\$

  where Q is charge (C) and t is time (s).

• Charge carriers – in metals the carriers are free electrons; in semiconductors they are electrons and holes.
• Potential difference (V) – energy transferred per unit charge:

  \$V=\frac{W}{Q}\qquad\text{(V)}\$

  where W is work (J).

• Power (P) – rate at which electrical energy is converted:

  \$P = VI = I^{2}R = \frac{V^{2}}{R}\qquad\text{(W)}\$

2. Resistance, resistivity & temperature‑coefficient (syllabus 9.3)

• Resistivity (ρ) – an intrinsic property of a material (Ω·m).

  For a uniform piece of length L and cross‑section A:

  \$R = \rho\frac{L}{A}\$

• Resistance (R) – the opposition to current of a particular component (Ω).

  It depends on ρ, geometry and temperature.

• Temperature‑coefficient of resistance (α) – expresses how R varies with temperature:

  \$R = R{0}\big[1+\alpha\,(T-T{0})\big]\$

  where R₀ is the resistance at reference temperature T₀.

2.1 Temperature‑coefficient for the three devices

2.2 Numerical example (metallic resistor)

Given a copper resistor with R₀ = 10 Ω at 20 °C and α = 4 × 10⁻³ °C⁻¹, the resistance at 70 °C is:

\$\$

R = 10\big[1+4\times10^{-3}(70-20)\big]

= 10\big[1+0.20\big] = 12\;\Omega

\$\$

The 20 % increase illustrates the positive temperature‑coefficient.

3. Practical circuit concepts (syllabus 10.1)

• e.m.f. (ε) – the ideal voltage supplied by a source when no current flows.
• Internal resistance (r) – real sources have a small series resistance; the terminal voltage is

  \$V = \varepsilon - Ir\$

• Potential divider – two series resistances R₁ and R₂ share the source voltage:

  \$\$V{R1}=V{\text{source}}\frac{R{1}}{R{1}+R{2}},\qquad

  V{R2}=V{\text{source}}\frac{R{2}}{R{1}+R{2}}\$\$

• Circuit symbols (official Cambridge symbols):

  • Metallic resistor –
  • Semiconductor diode –
  • Filament lamp –
  • Battery (ideal) –

4. Kirchhoff’s Voltage Law (KVL)

In any closed loop the algebraic sum of potential differences is zero:

\$\sum V = 0\$

For a series loop containing a battery, a metallic resistor, a diode and a filament lamp:

\$\varepsilon - Ir = V{\text{metal}} + V{\text{diode}} + V_{\text{lamp}}\$

Substituting the appropriate I‑V relationship for each element allows the loop current I to be solved.

5. I–V characteristics (the core of syllabus 9.3 & 10.2)

5.1 Metallic conductor at constant temperature

• Ohm’s law applies: V = IR (R is constant because T is fixed).
• The I–V graph is a straight line through the origin; slope = 1/R (or equivalently, the gradient is the conductance).

5.2 Semiconductor diode

• Reverse bias – only a tiny leakage (saturation) current flows: I ≈ ‑I_S (practically zero).
• Forward bias – current follows the Shockley equation:

  \$I = I_S\!\left(e^{\frac{qV}{kT}}-1\right)\$

  where I_S is the saturation current, q the elementary charge, k Boltzmann’s constant and T absolute temperature.

• The curve is almost flat for negative V, then rises sharply after the “knee” (≈ 0.6 V for Si, ≈ 0.3 V for Ge).

5.3 Filament lamp

• Resistance increases with temperature, so the I–V curve is concave upward.
• At low V the filament is cool → low R → steep initial slope (large dI/dV).
• As V rises the filament heats, R grows, and the slope diminishes, giving a curve that bends toward the horizontal.

6. Comparison of the three devices

7. Practical activity (Paper 3/5 skill)

Objective: Obtain the I‑V curves of a metal resistor and a filament lamp using a digital volt‑ammeter.

1. Construct the circuit: Battery (ε) → Ammeter → Device → Voltmeter (across the device) → Battery. Include the internal resistance symbol if required.
2. Vary the source voltage in small steps (e.g., 0 V → 12 V in 1 V increments) and record the corresponding current.
3. Plot I (y‑axis) against V (x‑axis):

   • Metal resistor: expect a straight line; calculate R from the slope (R = ΔV/ΔI).
   • Filament lamp: the curve should bow upward; determine the change in slope and discuss the role of the temperature‑coefficient.

4. Possible sources of error:

   • Contact resistance at the terminals.
   • Battery internal resistance (affects the terminal voltage).
   • Instrument tolerances (typically ±0.5 % for digital meters).
   • Heat loss from the filament to the surroundings.

5. Analyse the data by comparing the experimental resistance values with those predicted from the temperature‑coefficient formulas in section 2.

8. Quick sketching checklist (exam tip)

1. Mark the origin (0 V, 0 A) for every curve.
2. Metallic conductor – draw a straight line; label the slope as 1/R.
3. Diode – indicate:

   • Reverse‑bias leakage (tiny current),
   • Knee voltage (~0.6 V for Si),
   • Exponential rise in forward bias.

4. Filament lamp – start with a steep slope near the origin, then gradually flatten as V increases.
5. Annotate each graph with the relevant temperature‑coefficient (positive or negative) and the appropriate power formula where useful.

9. Syllabus coverage check