Physics 9702/51 — October/November 2017

Background Concept

An LDR is a resistor whose resistance depends on the light intensity (illuminance) falling on it: higher light intensity usually gives lower resistance. Here, the LDR is placed under water, and as the depth $d$ increases, less light reaches it (due to absorption/scattering and spreading), so the resistance changes.

The suggested model is

$$R = \frac{4\pi d^2}{K}$$

This has the form $R \propto d^2$ with proportionality constant $\frac{4\pi}{K}$. A standard way to test such a relationship is to keep everything else constant, vary $d$, measure $R$, and check whether a graph of $R$ against $d^2$ is a straight line.

To determine $K$, we use the idea that for a linear graph $y = mx + c$, the gradient $m$ can be measured and then rearranged to solve for the unknown constant.

Understanding the Question

You must design (plan) a workable experiment that:

• varies the depth $d$ of the LDR below the water surface,
• measures the resistance $R$ for each depth,
• controls other variables that affect LDR resistance (lamp brightness, ambient light, alignment, temperature, water clarity),
• analyses results in a way that tests the proposed equation,
• explains how the value of $K$ is obtained from the results,
• includes a labelled diagram of the set-up,
• states sensible safety precautions (water + electricity + hot lamp).

So the key deliverables are: method, measurements, controls, graph/analysis, and safety.

Approach

1. Choose a reliable way to measure $R$ while the LDR is illuminated. In a planning question, a potential divider is a strong method because it avoids using an ohmmeter mode (which can be sensitive to contact issues and may be awkward while the LDR is in water).
2. Vary $d$ systematically using a clamp and a ruler scale.
3. For each $d$, measure voltages and calculate $R$ using $R = V/I$.
4. Linearise: since $R \propto d^2$, plot $R$ vs $d^2$.
5. Use gradient to calculate $K$ via $m = 4\pi/K$.
6. Build in control measures and repeats to reduce uncertainty.

Step-by-Step Reasoning

1) Circuit to obtain $R$
Put the LDR in series with a known resistor $R_0$ across a fixed d.c. supply $V_s$. This makes a potential divider. Measure the voltage across $R_0$ (call it $V_0$) and across the LDR (call it $V_{\text{LDR}}$).

Because $R_0$ is known, you can find the current in the series circuit:

$$I = \frac{V_0}{R_0}$$

Then the resistance of the LDR at that depth is:

$$R = \frac{V_{\text{LDR}}}{I}$$

This is direct use of Ohm's law $R=V/I$ and is valid because the same current flows through series components.

2) Varying $d$ and measuring it
Mount the LDR on a clamp so it can be moved vertically. Measure $d$ as the vertical distance from the water surface to the active face of the LDR. Use a ruler fixed to the tank, or measure from a reference mark.

Choose a range of at least 6 depths (more is better), spaced so that $d^2$ covers a good range (this improves the reliability of the gradient).

3) Controlling variables (why it matters)
The LDR resistance is affected by more than depth:

• Lamp brightness: must be constant, so use a stable power supply and do not move the lamp.
• Distance above water: keep lamp-to-surface distance constant so only $d$ changes.
• Alignment: keep the LDR directly under the lamp each time; otherwise changing the angle changes the intensity at the LDR.
• Ambient light: stray light adds to the illumination and changes $R$; use a shield/darkened area.
• Water clarity and bubbles: scattering/absorption changes with murkiness and bubbles; use the same clear water and remove bubbles.
• Temperature: resistances can drift with temperature; allow lamp to stabilise, take readings reasonably quickly, and keep conditions similar.

4) Analysis to test the relationship
From the model:

$$R = \frac{4\pi}{K} d^2$$

Let $x = d^2$ and $y = R$. Then:

$$y = \left(\frac{4\pi}{K}\right) x$$

So if the relationship is correct:

• a plot of $R$ (y-axis) against $d^2$ (x-axis) should give a straight line,
• the intercept should be close to zero (within experimental uncertainty).

5) Determining $K$ from the graph
The gradient is

$$m = \frac{\Delta R}{\Delta(d^2)} = \frac{4\pi}{K}$$

so

$$K = \frac{4\pi}{m}.$$

You would calculate $m$ from a large triangle on the best-fit line. If the question expects uncertainty discussion, you could repeat readings and/or draw worst acceptable lines to estimate an uncertainty in $m$, then propagate it to $K$ (since $K \propto 1/m$, the percentage uncertainty in $K$ is approximately the same as that in $m$).

6) Safety
The main hazards are water near electrical equipment and a hot lamp. Control them by using low voltage, keeping meters and connections away from spills, drying hands, and ensuring the lamp is secure and not touched when hot.

Key Takeaways

• A planning question is scored on: clear variables, a workable measurement method, control of variables, and a correct graphing method to test the law.
• To test $R \propto d^2$, plot $R$ vs $d^2$ and look for a straight line.
• Use the gradient to find constants: if $m = 4\pi/K$, then $K = 4\pi/m$.

Common Mistakes

• Plotting $R$ against $d$ instead of against $d^2$ (would not be linear if the model is correct).
• Not stating how $R$ is measured (must explain what you measure and how you calculate $R$).
• Failing to control lamp brightness/position or ambient light, which can dominate the effect of $d$.
• Measuring $d$ inconsistently (e.g. from the bottom of the beaker sometimes and from the surface at other times).
• No mention of repeats/averaging and no attempt to obtain a best-fit line.

Things to Be Careful About

• Keep units consistent: $d$ in $\text{m}$ gives $d^2$ in $\text{m}^2$; then the gradient has units $\Omega\,\text{m}^{-2}$, so $K$ has units consistent with the given equation.
• Ensure the LDR is waterproofed without blocking light (use a thin transparent covering).
• Avoid parallax when reading $d$; read the meniscus at eye level.
• Let the water surface settle before reading depth and taking voltage readings.
• When finding gradient, use two well-separated points on the best-fit line (not just two adjacent plotted points).