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1. Diagram & Experimental Layout

Draw a shallow insulating tray containing the conducting solution of depth \(h\). Two vertical metal rods of diameter \(D\) are immersed in the solution. These rods are kept parallel and separated by a distance \(L\). The rods are connected in a circuit with a low-voltage alternating current (AC) power supply, an ammeter in series, and a voltmeter in parallel across the rods.

2. Identification of Variables

• Independent variable: Distance between the parallel rods \(L\).
• Dependent variable: Resistance \(R\) between the rods (calculated from \(R = V/I\)).
• Controlled variables: Depth of the conducting liquid \(h\), diameter of the rods \(D\), concentration of the conducting liquid/solution, and temperature of the liquid.

3. Method of Data Collection

• Measure the diameter \(D\) of the metal rods using a micrometer screw gauge. Measure at different points along the rod and average the readings.
• Fill the shallow tray with the conducting solution. Measure the depth \(h\) of the liquid using a micrometer depth gauge, a traveling microscope, or a vertical needle attached to a vernier scale.
• Place the rods vertically in the solution. Use a ruler to measure the separation \(L\) between the centers of the rods. Ensure they are parallel by measuring the separation at the top and the bottom.
• Switch on the AC power supply. Record the potential difference \(V\) from the voltmeter and the current \(I\) from the ammeter. Calculate the resistance using \(R = \frac{V}{I}\).
• Change the distance \(L\) and repeat the measurements of \(V\) and \(I\) for at least six different values of \(L\).

4. Method of Analysis

• The equation is given by: \(R = \frac{\rho}{\pi h} \ln\left(\frac{2L}{D}\right)\)
• Plot a graph of \(R\) on the y-axis against \(\ln\left(\frac{2L}{D}\right)\) on the x-axis.
• If the relationship is valid, the graph should be a straight line passing through the origin.
• The gradient \(m\) of this straight line is given by: \(m = \frac{\rho}{\pi h}\)
• Therefore, the resistivity \(\rho\) can be determined using: \(\rho = m \times \pi h\)

5. Safety & Quality Details

• Safety: Use a low-voltage power supply. Switch off the power circuit when assembling or adjusting the position of the rods to prevent electric shock. Wipe up any liquid spills immediately to prevent slip hazards and short circuits.
• Quality of results: Use an AC supply instead of a DC supply to prevent polarization at the electrodes (electrolysis) which would build up gas bubbles and change the apparent resistance. Keep the solution well-stirred and at a constant temperature (monitored with a thermometer) because resistivity depends strongly on temperature. Ensure the tray is perfectly horizontal on the workbench to maintain a uniform depth \(h\) across the entire setup.

(a) Calculated Table Values

The values of \(T^2\) and their absolute uncertainties \(\Delta(T^2)\) are calculated using \(T^2\) and \(\Delta(T^2) \approx 2 T \Delta T\) (or the max/min method):

• For \(h = 5.0\text{ cm}\): \(T^2 = 0.46^2 = 0.21\text{ s}^2\), \(\Delta(T^2) = 2(0.46)(0.02) = 0.018 \approx 0.02\text{ s}^2\)
• For \(h = 8.0\text{ cm}\): \(T^2 = 0.58^2 = 0.34\text{ s}^2\), \(\Delta(T^2) = 2(0.58)(0.02) = 0.023 \approx 0.02\text{ s}^2\)
• For \(h = 11.0\text{ cm}\): \(T^2 = 0.68^2 = 0.46\text{ s}^2\), \(\Delta(T^2) = 2(0.68)(0.02) = 0.027 \approx 0.03\text{ s}^2\)
• For \(h = 14.0\text{ cm}\): \(T^2 = 0.76^2 = 0.58\text{ s}^2\), \(\Delta(T^2) = 2(0.76)(0.02) = 0.030 \approx 0.03\text{ s}^2\)
• For \(h = 17.0\text{ cm}\): \(T^2 = 0.84^2 = 0.71\text{ s}^2\), \(\Delta(T^2) = 2(0.84)(0.02) = 0.034 \approx 0.03\text{ s}^2\)
• For \(h = 20.0\text{ cm}\): \(T^2 = 0.91^2 = 0.83\text{ s}^2\), \(\Delta(T^2) = 2(0.91)(0.02) = 0.036 \approx 0.04\text{ s}^2\)

The completed table is:
\(\begin{array}{|c|c|c|} \hline h / \text{cm} & T / \text{s} & T^2 / \text{s}^2 \\ \hline 5.0 & 0.46 \pm 0.02 & 0.21 \pm 0.02 \\ \hline 8.0 & 0.58 \pm 0.02 & 0.34 \pm 0.02 \\ \hline 11.0 & 0.68 \pm 0.02 & 0.46 \pm 0.03 \\ \hline 14.0 & 0.76 \pm 0.02 & 0.58 \pm 0.03 \\ \hline 17.0 & 0.84 \pm 0.02 & 0.71 \pm 0.03 \\ \hline 20.0 & 0.91 \pm 0.02 & 0.83 \pm 0.04 \\ \hline \end{array}\)

(b) Graph Plotting

Plot \(T^2\) on the vertical axis against \(h\) on the horizontal axis. Draw vertical error bars of length equal to \(\pm \Delta(T^2)\) for each point. The best-fit line is a straight line passing through the points. The worst acceptable straight line is the steepest or shallowest possible line that passes through all of the error bars.

(c) Gradient and y-Intercept

• Best-Fit Line Gradient: Using coordinates on the line (5.0, 0.21) and (20.0, 0.83):
  \(m_{\text{best}} = \frac{0.83 - 0.21}{20.0 - 5.0} = 0.0413\text{ s}^2\text{ cm}^{-1}\)
• Worst Acceptable Line Gradient: Connecting the bottom of the first error bar (5.0, 0.19) to the top of the last error bar (20.0, 0.87):
  \(m_{\text{worst}} = \frac{0.87 - 0.19}{20.0 - 5.0} = 0.0453\text{ s}^2\text{ cm}^{-1}\)
• Uncertainty in Gradient: \(\Delta m = |m_{\text{best}} - m_{\text{worst}}| = |0.0413 - 0.0453| = 0.0040\text{ s}^2\text{ cm}^{-1}\)
• Best-Fit Line Intercept:
  \(c_{\text{best}} = 0.21 - (0.0413 \times 5.0) = 0.0035 \approx 0.004\text{ s}^2\)
• Worst Acceptable Line Intercept:
  \(c_{\text{worst}} = 0.19 - (0.0453 \times 5.0) = -0.0365\text{ s}^2\)
• Uncertainty in Intercept: \(\Delta c = |c_{\text{best}} - c_{\text{worst}}| = |0.0035 - (-0.0365)| = 0.040\text{ s}^2\)

Hence:
Gradient = \(0.0413 \pm 0.0040\text{ s}^2\text{ cm}^{-1}\) (or \(4.13 \pm 0.40\text{ s}^2\text{ m}^{-1}\))
Intercept = \(0.004 \pm 0.040\text{ s}^2\)

(d) Determining Constants \(a\) and \(b\)

• Comparing \(T^2 = a h + b\) with \(y = mx + c\), we see that:
  \(a = \text{gradient}\)
  \(b = \text{y-intercept}\)
• Using SI units (converting \(h\) to meters):
  \(a = 4.13 \pm 0.40\text{ s}^2\text{ m}^{-1}\)
  \(b = 0.004 \pm 0.040\text{ s}^2\)