recall and use the principle of the potentiometer as a means of comparing potential differences

Potentiometer – Comparing Potential Differences (Cambridge AS/A‑Level Physics 9702)

Learning Objective

Recall and apply the principle of the potentiometer as a null‑method technique for comparing potential differences (emf’s) without drawing current from the source.

Syllabus Context – Topic Map

Definitions (quick‑reference box)

Key Concept Checklist

• Uniform resistive wire → constant potential gradient \(k = V_{\text{total}}/L\).
• Driver internal resistance \(r_d\) modifies the gradient; it can be measured and corrected.
• Kirchhoff’s first law (current law) for series networks → potential‑divider equation.
• Null‑method: at balance no current is drawn from the unknown source, eliminating loading error.
• Temperature coefficient of the wire and contact resistance affect the gradient.
• Propagation of uncertainties for a single emf and for ratios of emfs.

1. The Potentiometer Principle

A potentiometer consists of a straight length L of uniform resistive wire (resistance \(R_{\text{wire}}\)) connected to a stable driver (voltage source). When a current I flows, the voltage drop along the wire is linear:

\[

V(x)=k\,x,\qquad k=\frac{V{\text{total}}}{L}=\frac{I\,(R{\text{wire}}+r_d)}{L}

\]

• Zero‑potential end: connected to the negative terminal of the driver.
• Positive end: connected to the positive terminal.
• Because the wire is uniform, k (the potential gradient) is constant provided the temperature is uniform and the driver current is steady.

Driver internal resistance – measurement and correction

1. Set the driver to a known voltage \(V_d\) (read from a calibrated voltmeter).
2. Measure the total resistance of the driver (including its internal resistance) with a multimeter: \(R{\text{driver}} = rd +\) any external series resistance you deliberately add.
3. Measure the resistance of the potentiometer wire, \(R_{\text{wire}}\), by disconnecting it from the driver and using the multimeter.
4. The true gradient is then

   \[

   k = \frac{Vd}{L}\;\frac{R{\text{wire}}+rd}{R{\text{wire}}}

   \]

   The factor \(\frac{R{\text{wire}}+rd}{R_{\text{wire}}}\) corrects for the voltage drop across the driver’s internal resistance.

2. Connection to the Potential‑Divider Concept

The uniform wire can be regarded as an infinite series of infinitesimal resistors \(dR\). Over a length \(x\) the equivalent resistance is \(\frac{R_{\text{wire}}}{L}\,x\). Substituting this into the divider formula gives

\[

V(x)=V_{\text{total}}\frac{x}{L}=k\,x,

\]

which is exactly the potentiometer relation derived above. The key advantage of the potentiometer is that the “load” (the unknown source) is connected only at the balance point, where the galvanometer reads zero, so the divider is never loaded – a point worth emphasising for AO3 (explain why the method is superior to a simple voltage‑divider measurement).

3. Practical Implementation – Checklist & Wiring Diagram

3.1 Practical Checklist (Paper 3/5)

1. Safety & preparation

   • Check that the driver voltage is within the rating of the wire and the galvanometer.
   • Ensure the wire is straight, insulated and clamped to prevent movement.
   • Verify that the sliding contact (knife‑edge) is clean, smooth and exerts light, consistent pressure.

2. Driver set‑up

   • Connect the driver positive terminal to the right‑hand end of the wire, negative to the left‑hand end.
   • Insert a known series resistor (if required) to set the desired current.
   • Measure and record the driver voltage \(Vd\) and the driver resistance \(rd\).

3. Galvanometer zeroing

   • Short the galvanometer leads, adjust the zero‑adjustment knob until the needle reads zero.
   • Reconnect the leads to the unknown source (see diagram).

4. Balance‑length measurement

   • Place the sliding contact near the zero‑potential end.
   • Slide it slowly until the galvanometer shows zero deflection (null point).
   • Read the length from the scale (to the nearest 0.1 mm) and record it.
   • Repeat the measurement at least three times and take the mean.

5. Temperature monitoring

   • Record the ambient temperature; if the wire is heated by the driver, allow it to stabilise before taking readings.

6. Data recording

   • Use the table in Section 7 (Data‑record template) to keep the format consistent with the assessment objectives.

3.2 Wiring Diagram (ASCII for quick reference)

+———[Driver +]———+———[R_wire]———+———[Driver –]———-
|               |
|               |
|               |
[Sliding]          [Galvanometer]
Contact            (zeroed)
|               |
+———[Unknown emf]———+

When the sliding contact is at the balance point the galvanometer reads zero; the potential difference across the unknown emf equals the potential drop along the wire between the zero end and the contact.

4. Step‑by‑Step Procedure for Comparing Two Unknown EMFs

1. Calibrate the gradient k using a standard cell of known emf \(E_{\text{ref}}\) (or directly from the measured driver voltage):

   \[

   k = \frac{E{\text{ref}}}{l{\text{ref}}}\quad\text{or}\quad

   k = \frac{Vd}{L}\frac{R{\text{wire}}+rd}{R{\text{wire}}}.

   \]

2. Connect the first unknown cell \(E1\) in series with the galvanometer and the sliding contact. Slide until null and record the balance length \(l1\).
3. Repeat for the second unknown cell \(E2\) to obtain \(l2\).
4. Calculate the emfs:

   \[

   E1 = k\,l1,\qquad E2 = k\,l2,\qquad

   \frac{E1}{E2}= \frac{l1}{l2}.

   \]

5. Propagate uncertainties (see Section 6) and present the final results with appropriate significant figures.

5. Uncertainty and Error Analysis

For a single emf:

\[

\Delta E = \sqrt{(k\,\Delta l)^2 + (l\,\Delta k)^2}

\]

where

• \(\Delta l\) – uncertainty of the balance length (typically ±0.05 mm).
• \(\Delta k\) – combined uncertainty from driver voltage, wire length and driver resistance:

  \[

  \Delta k = k\sqrt{\left(\frac{\Delta Vd}{Vd}\right)^2

  +\left(\frac{\Delta L}{L}\right)^2

  +\left(\frac{\Delta rd}{R{\text{wire}}+r_d}\right)^2 }.

  \]

For the ratio of two emfs:

\[

\frac{\Delta (E1/E2)}{E1/E2}= \sqrt{\left(\frac{\Delta l1}{l1}\right)^2+\left(\frac{\Delta l2}{l2}\right)^2}

\]

Typical classroom values:

• Length \(L\) measured with a metre rule: \(\Delta L = \pm0.5\) mm.
• Driver voltage from a regulated supply: \(\Delta V_d = \pm0.01\) V.
• Driver internal resistance from a multimeter: \(\Delta r_d = \pm0.01\) Ω.
• Temperature gradient (if not insulated): add a systematic term of up to ±1 % of \(k\).

6. Worked Examples

Example 1 – Basic potentiometer measurement (no driver resistance)

Given:

• L = 1.00 m
• Driver voltage \(V_d = 5.00\) V (stable)
• Balance length for the unknown cell = 0.320 m
• \(\Delta l = \pm0.05\) mm

Solution:

\[

k = \frac{V_d}{L}=5.00\ \text{V m}^{-1}

\]

\[

E = k\,l = 5.00 \times 0.320 = 1.60\ \text{V}

\]

\[

\Delta E = k\,\Delta l = 5.00 \times 0.00005 = 2.5\times10^{-4}\ \text{V}\approx\pm0.0003\ \text{V}

\]

Example 2 – Including driver internal resistance

Data:

• L = 1.20 m
• Driver voltage \(V_d = 6.00\) V
• Driver internal resistance \(r_d = 0.20\) Ω
• Wire resistance \(R_{\text{wire}} = 10.0\) Ω
• Balance length \(l = 0.450\) m
• \(\Delta Vd = \pm0.01\) V, \(\Delta rd = \pm0.01\) Ω, \(\Delta l = \pm0.05\) mm

Gradient (uncorrected): \(k0 = Vd/L = 5.00\ \text{V m}^{-1}\).

Corrected gradient:

\[

k = k0\frac{R{\text{wire}}+rd}{R{\text{wire}}}=5.00\frac{10.0+0.20}{10.0}=5.10\ \text{V m}^{-1}

\]

EMF:

\[

E = k\,l = 5.10 \times 0.450 = 2.30\ \text{V}

\]

Uncertainty:

\[

\Delta k = k\sqrt{\left(\frac{0.01}{6.00}\right)^2+\left(\frac{0.01}{10.20}\right)^2}=0.009\ \text{V m}^{-1}

\]

\[

\Delta E = \sqrt{(k\Delta l)^2+(l\Delta k)^2}

= \sqrt{(5.10\times0.00005)^2+(0.450\times0.009)^2}

\approx 0.004\ \text{V}

\]

\[

\boxed{E = 2.30 \pm 0.004\ \text{V}}

\]

Example 3 – Demonstrating the advantage of the null method (loading effect)

Suppose a simple voltage divider is used to measure an unknown emf \(E\) of 2.00 V. The divider output is connected to a digital voltmeter of internal resistance 10 MΩ. If the divider has a total resistance of 1 MΩ, the voltmeter draws a current that reduces the measured voltage by about 0.2 %.

Using the potentiometer, the galvanometer is set to zero; no current flows from the unknown source, so the measured emf is free from this loading error. This conceptual point should be highlighted in the answer to an AO3 question.

7. Data‑Record Template (Paper 3/5)

8. Common Pitfalls & Mitigation Strategies

• Driver drift: Use a regulated DC supply; re‑measure the balance length after any temperature change.
• Contact resistance: Keep the sliding contact clean, use a light but firm pressure, and replace worn contacts.
• Non‑uniform wire: Verify straightness, secure the wire, and insulate it from drafts; a water‑bath enclosure can minimise temperature gradients.
• Reading from the wrong end: Clearly label the zero‑potential end on the apparatus and on the worksheet.
• Insufficient galvanometer sensitivity: Use a galvanometer with a sensitivity of at least 10 µA div⁻¹ or a digital null detector.
• Systematic error from temperature: Record ambient temperature; if the wire heats noticeably, allow it to stabilise or apply a temperature‑coefficient correction.

9. Summary

• The potentiometer is a high‑precision null‑method instrument for comparing emfs without drawing current from the source.
• It relies on a uniform potential gradient along a calibrated resistive wire; the gradient can be corrected for driver internal resistance.
• Because the unknown source is only connected at the balance point, loading effects are eliminated – a key point for AO3 explanations.
• Accurate results require careful calibration, temperature control, and proper uncertainty propagation.

10. What’s Next?

Having mastered the potentiometer, you will encounter the following related topics in the remaining syllabus:

• Electric fields (Section 18) – understanding the uniform field along the wire that gives rise to the constant gradient.
• Capacitance (Section 19) – using potentiometers in RC‑timing experiments and for measuring dielectric constants.
• Potential dividers (Section 10.2) – designing calibrated voltage dividers and exploring loading effects in contrast to the null method.
• Energy considerations – relating the work done in moving charge along the wire to the measured emf.

11. Symbol Table