explain the use of thermistors and light-dependent resistors in potential dividers to provide a potential difference that is dependent on temperature and light intensity

Potential Dividers (Voltage Dividers)

Potential dividers are a fundamental part of the Cambridge International AS & A Level Physics syllabus (Section 10.3). They provide a known fraction of an input voltage and are used for:

• Measuring an unknown voltage (potentiometer + null method).
• Providing a bias voltage for active devices.
• Converting a physical quantity (temperature, light) into a voltage using a sensor element such as a thermistor or a light‑dependent resistor (LDR).

1. Potential‑Divider Rule – Derivation from Kirchhoff’s Voltage Law

Consider two resistors, R₁ and R₂, in series across a stable supply V_in. The current I is the same through both resistors:

\[

I=\frac{V{\text{in}}}{R{1}+R{2}} \qquad\text{(by KVL: }V{\text{in}}=IR{1}+IR{2}\text{)}

\]

The voltage across R₂ (the output voltage) is:

\[

V{\text{out}} = I R{2}=V{\text{in}}\frac{R{2}}{R{1}+R{2}}

\]

This expression is known as the potential‑divider rule. It is valid for any pair of series resistances, whether they are fixed, variable, or a mixture of both.

Design considerations

• Fixed resistor (R_fixed) – choose a precision resistor with a low temperature coefficient (TC) so its value remains essentially constant.
• Total resistance – should allow a safe current from the supply (avoid overheating the sensor).
• Output impedance –

  \[

  R{\text{out}}=\frac{R{1}R{2}}{R{1}+R_{2}}

  \]

  This must be much lower than the input impedance of the following stage (e.g., an ADC).

• When a sensor element provides the variable resistance, the fixed resistor supplies the reference value.

2. Potentiometer – Null‑Method Voltage Comparator

A potentiometer is a long, uniform‑resistance wire with a sliding contact. It implements an extremely accurate potential divider because the ratio of the two wire sections is set mechanically.

Balance (null) condition

Let the unknown voltage be V_u. The potentiometer is adjusted until a galvanometer connected between the unknown source and the tap shows zero current. At balance:

\[

V{u}=V{\text{ref}}\;\frac{R{\text{tap}}}{R{\text{total}}}

\]

where V_ref is a known reference voltage applied across the whole wire, R_tap is the resistance of the wire segment between the left end and the tap, and R_total is the total wire resistance. Because the galvanometer carries no current, the measurement does not load the unknown source, giving very high accuracy.

Practical steps

1. Connect the known reference voltage across the potentiometer ends.
2. Connect the unknown voltage source and a sensitive galvanometer between the tap and the reference end.
3. Slide the tap until the galvanometer reads zero (null).
4. Read the length (or resistance) of the wire segment; use the proportion above to calculate V_u.

3. Galvanometer Requirements in Null Methods

• Internal resistance \(R_{g}\) should be at least ten times larger than the total resistance of the potentiometer circuit (typically > 10 MΩ) so that the galvanometer does not disturb the balance.
• It must be sufficiently sensitive to detect micro‑ampere currents, ensuring a true null can be observed.

4. Thermistors in Potential Dividers

A thermistor is a semiconductor resistor whose resistance varies strongly with temperature.

Types

• NTC (Negative Temperature Coefficient) – resistance falls as temperature rises.
• PTC (Positive Temperature Coefficient) – resistance rises as temperature rises.

Linear approximation (limited range)

\[

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

\]

• \(R{0}\) – resistance at reference temperature \(T{0}\) (usually 25 °C).
• \(\alpha\) – temperature coefficient (negative for NTC, positive for PTC).

Using the thermistor as the lower resistor (\(R{2}=R{T}\))

\[

V{\text{out}}(T)=V{\text{in}}\;\frac{R{T}}{R{\text{fixed}}+R_{T}}

\]

Design example – NTC thermistor

Resistances:

\[

\begin{aligned}

R_{20} &= 10\,\text{k}\Omega\bigl[1-3.9\times10^{-3}(20-25)\bigr]=10.195\ \text{k}\Omega,\\[4pt]

R_{30} &= 10\,\text{k}\Omega\bigl[1-3.9\times10^{-3}(30-25)\bigr]=9.805\ \text{k}\Omega.

\end{aligned}

\]

Corresponding output voltages:

\[

\begin{aligned}

V_{\text{out}}(20^{\circ}\text{C}) &= 5\;\frac{10.195}{10+10.195}=2.55\ \text{V},\\[4pt]

V_{\text{out}}(30^{\circ}\text{C}) &= 5\;\frac{9.805}{10+9.805}=2.48\ \text{V}.

\end{aligned}

\]

The ≈ 70 mV change can be fed to an ADC or amplified for temperature monitoring.

Thermistor data summary

5. Light‑Dependent Resistors (LDRs) in Potential Dividers

An LDR (photoresistor) changes its resistance with incident illuminance \(E\) (lux). The characteristic is non‑linear and can be approximated by:

\[

R_{L}=K\,E^{-\gamma}

\]

• \(K\) and \(\gamma\) are material‑dependent constants (\(\gamma\approx0.5\!-\!0.7\)).

Using the LDR as the lower resistor (\(R{2}=R{L}\))

\[

V{\text{out}}(E)=V{\text{in}}\;\frac{K\,E^{-\gamma}}{R_{\text{fixed}}+K\,E^{-\gamma}}

\]

Because \(R_{L}\) falls when light intensity rises, the output voltage increases as the illumination decreases (when the LDR is the lower resistor).

Design example – LDR

\[

\begin{aligned}

V_{\text{out}}(\text{dark}) &= 12\;\frac{100}{5+100}=11.4\ \text{V},\\[4pt]

V_{\text{out}}(\text{bright}) &= 12\;\frac{2}{5+2}=3.43\ \text{V}.

\end{aligned}

\]

The large swing makes the LDR‑divider suitable for ambient‑light detection in robotics, automatic lighting, and safety systems.

LDR data summary

6. Practical Tips for Sensor‑Based Dividers

• Use a precision, low‑TC fixed resistor so that variations in output voltage are dominated by the sensor.
• Place the divider as close as possible to the sensor to minimise lead resistance and stray capacitance.
• If the sensor resistance spans several decades, consider a two‑stage divider or a logarithmic amplifier to keep the output within the linear range of the measuring instrument.
• For very high‑value LDRs or thermistors, guard against leakage currents by:

  • Employing a clean PCB layout with guard rings.
  • Using high‑impedance, low‑bias‑current input stages (e.g., instrumentation amplifiers).

• When using the null method, verify that the galvanometer resistance is much greater than the total divider resistance so that the balance is not disturbed.

7. Summary Table – Sensor‑Based Potential Dividers