explain that the resistance of a filament lamp increases as current increases because its temperature increases

Cambridge A‑Level Physics (9702) – Resistance, Resistivity and Filament Lamps

1. Electric Current (Syllabus 9.1)

• Current, I – rate of flow of charge

  \[

  I=\frac{\Delta Q}{\Delta t}\;(\text{A})

  \]

• In a metallic conductor the drift of free electrons gives

  \[

  I=nqAv

  \]

  • \(n\) – number density of charge carriers (m\(^{-3}\))
  • \(q=1.6\times10^{-19}\) C – charge of an electron
  • \(A\) – cross‑sectional area of the conductor (m\(^2\))
  • \(v\) – average drift speed (m s\(^{-1}\))

• Current direction is defined as the direction a positive charge would move (conventional current).

2. Potential Difference, EMF and Power (Syllabus 9.2)

• Potential difference (voltage) between two points

  \[

  V=\frac{W}{Q}\;(\text{V})

  \]

• Electromotive force (e.m.f.) – energy supplied per coulomb by a source when no current flows.
• Ohm’s law for a resistor (or a short segment of a conductor at a fixed temperature)

  \[

  V=IR

  \]

• Electrical power

  • \(P = VI\)
  • Using Ohm’s law: \(P = I^{2}R = \dfrac{V^{2}}{R}\)

3. Resistance and Resistivity (Syllabus 9.3)

3.1 Definition of resistance

Resistance \(R\) quantifies how strongly a material opposes the flow of electric current. SI unit: the ohm (Ω).

3.2 Resistivity – an intrinsic property

The resistivity \(\rho\) depends only on the material and its temperature. The geometric relation is

\[

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

\]

• \(L\) – length of the conductor (m)
• \(A\) – cross‑sectional area (m\(^2\))

3.3 Worked numeric example (copper wire)

3.4 Temperature dependence of resistance

For most metals the resistivity rises approximately linearly with temperature over a limited range:

\[

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

\tag{1}

\]

• \(R{0}\) – resistance at reference temperature \(T{0}\) (usually 20 °C).
• \(\alpha\) – temperature coefficient of resistance (K\(^{-1}\)). Positive for metals (e.g. copper \(\alpha = 3.9\times10^{-3}\) K\(^{-1}\)).
• Equation (1) is accurate only up to ≈ 500 °C; above this the relationship becomes noticeably non‑linear.

3.5 Derivation of the linear form

Starting from the definition \(\rho(T)=\rho{0}[1+\beta(T-T{0})]\) and substituting into \(R=\rho L/A\) gives

\[

R(T)=\frac{L}{A}\,\rho{0}[1+\beta(T-T{0})]

=R{0}[1+\beta(T-T{0})],

\]

where \(\beta\) is the material’s resistivity temperature coefficient. For metals \(\beta\approx\alpha\), yielding equation (1).

3.6 Non‑linear resistivity of tungsten (filament material)

Filament lamps operate at temperatures 2000 – 3000 °C, far beyond the linear range. The table below shows measured resistivity values for tungsten (source: NIST).

A graph of \(\rho\) versus \(T\) (placeholder) would clearly show the curvature that makes the filament a non‑ohmic device.

4. Why a Filament Lamp’s Resistance Increases with Current

1. Applying a voltage makes a current \(I\) flow through the filament.
2. The filament converts electrical energy into heat:

   \[

   P = I^{2}R\quad\text{(instantaneous power)}

   \]

3. Because tungsten has a large temperature coefficient (\(\alpha\approx4.5\times10^{-3}\) K\(^{-1}\)), the dissipated power raises its temperature dramatically.
4. The rise in temperature increases the resistivity according to the data in §3.6, so the resistance \(R\) grows.
5. Thus, as the current increases, the filament becomes hotter, its resistivity climbs, and the measured resistance is larger – the lamp is a non‑ohmic component.

4.1 Quantitative example – tungsten filament

Assume a filament of length \(L=5\;\text{mm}\) and cross‑sectional area \(A=0.02\;\text{mm}^{2}\). Using the resistivity values from the table:

The resistance more than quintupled as the filament heated from room temperature to its normal operating temperature (~2500 °C).

4.2 Error‑analysis note

• Using the linear model (1) at 2500 °C would underestimate the resistance by ≈ 35 %.
• When solving circuit problems that involve a filament lamp, always adopt either the tabulated \(\rho(T)\) data or the measured V‑I characteristic supplied in the exam question.

5. Circuit Symbols (Syllabus 10.1)

Standard symbols used in Cambridge A‑Level examinations:

	Battery / e.m.f. source
	Resistor (fixed)
	Filament lamp (non‑ohmic)
	Ammeter (connected in series)
	Voltmeter (connected in parallel)
	Switch
	Rheostat / variable resistor

6. Potential‑Divider Using a Filament Lamp (Syllabus 10.3)

A filament lamp can act as the variable resistor in a potential‑divider. The circuit is shown in Figure 1.

Given a 12 V supply, \(R_{1}=10\;\Omega\) and a lamp whose resistance at operating temperature is ≈ 80 Ω, the voltage across the lamp is

\[

V{L}=V{\text{s}}\frac{R{L}}{R{1}+R_{L}}

=12\;\text{V}\times\frac{80}{10+80}=10.7\;\text{V}.

\]

If the lamp cools (e.g., by reducing the supply voltage), its resistance falls, changing the division ratio. This illustrates the practical use of a non‑ohmic component in a voltage‑divider and links syllabus sections 9.3 and 10.3.

7. Kirchhoff’s Laws with a Filament Lamp (Syllabus 10.2)

Consider the circuit of Figure 2: a 12 V battery supplies two parallel branches – one contains a filament lamp (resistance \(R{L}(T)\)), the other a fixed resistor \(R{1}=10\;\Omega\). Both branches re‑join before a series resistor \(R_{2}=5\;\Omega\) returns to the battery.

Loop rule (clockwise):

\[

12\;\text{V} - I{2}R{2} - I{1}R{1} - V_{L}=0

\tag{2}

\]

Junction rule at the node where the branches split:

\[

I{2}=I{1}+I_{L}

\tag{3}

\]

Because \(V{L}=I{L}R{L}(T)\) and \(R{L}\) depends on the instantaneous temperature, equations (2)–(3) must be solved iteratively (or graphically using the lamp’s V‑I curve). This demonstrates how a temperature‑dependent resistance influences current distribution.

8. Practical Circuit Considerations (Syllabus 10.4)

• Current‑limiting resistor or rheostat – placed in series with the lamp to prevent runaway heating when the supply is switched on.
• Measuring the V‑I characteristic – connect a voltmeter across the lamp and an ammeter in series; vary the supply voltage in small steps and record the readings.
• Safety – the filament can reach >2500 °C; use insulated leads, avoid touching the bulb, and allow it to cool before handling.
• Calibration of the temperature‑coefficient – by measuring resistance at two known temperatures (e.g., room temperature and after a known heating period) and applying equation (1) to estimate \(\alpha\).

9. Summary

The resistance of a filament lamp rises with current because the electrical power dissipated as heat raises the filament’s temperature. Tungsten’s resistivity increases sharply with temperature; the simple linear model \(R=R{0}[1+\alpha(T-T{0})]\) is only valid up to ≈ 500 °C, so for a lamp operating at ≈ 2500 °C the non‑linear \(\rho(T)\) data must be used. This temperature‑dependent behaviour makes the lamp a non‑ohmic component, requiring V‑I curves, iterative calculations, or graphical methods when analysing circuits. Mastery of these ideas is essential for AO1 (knowledge), AO2 (application) and AO3 (analysis/evaluation) in the Cambridge A‑Level Physics syllabus.

10. Links to Assessment Objectives

• AO1 – Knowledge and Understanding: definitions of current, potential difference, resistance, resistivity; linear and non‑linear temperature‑coefficient relations; physical reason for heating in a filament.
• AO2 – Application: use of \(R=\rho L/A\), equation (1), tabulated \(\rho(T)\) for tungsten; solving circuits that contain a filament lamp using Kirchhoff’s laws, potential‑divider analysis, and current‑limiting design.
• AO3 – Analysis and Evaluation: interpreting the curved V‑I characteristic; assessing the limits of the linear model; error analysis when approximating with equation (1); evaluating iterative vs. graphical solution methods for circuits with temperature‑dependent resistance.