Figure 1 shows apparatus used to investigate the inverse-square law for gamma radiation - AQA - A-Level Physics - Question 1 - 2021 - Paper 3

C_b = 630 counts in 15 minutes, therefore:

1. Convert the time to seconds: 15 minutes = 900 seconds.

2. Calculate the background count rate R_b:

   $R_b = \frac{C_b}{t} = \frac{630}{900} = 0.7 \text{ counts/s}$

Calculate R_c for d = 530 mm:

1. Use the recorded counts:

   $R_1 = \frac{90}{100} = 0.9 \text{ counts/s}$ $R_2 = \frac{117}{100} = 1.17 \text{ counts/s}$ $R_3 = \frac{102}{100} = 1.02 \text{ counts/s}$

2. Calculate the mean count rate without background:

   $R_{mean} = \frac{R_1 + R_2 + R_3}{3} = \frac{0.9 + 1.17 + 1.02}{3} = 1.03 \text{ counts/s}$

3. Adjust for background:

   $R_c = R_{mean} - R_b = 1.03 - 0.7 = 0.33 \text{ counts/s}$

4. For d = 380 mm:

   $R_c = 0.76 \text{ counts/s}$

Since R_c is proportional to 1/d^2:

$\frac{R_{c(d=530)}}{R_{c(d=380)}} \approx \frac{0.33}{0.76} \text{ supports } k/d^2$ thus confirming that the data follows the inverse-square law.

To safely reduce the value of d:

1. Lower the detector while ensuring it remains stable to avoid any accidents. Clamps should be adjusted to prevent movement during measurements.

2. A set-square can also be used to verify vertical alignment as the distance is reduced.

3. Ensure no one stands directly in the line of the radiation source during adjustments.

This procedure reduces the distance while minimizing exposure risks and ensures accurate collection of data.

To determine Δd:

1. Collect sufficient data points from multiple measurements of d.
2. Calculate the difference in d values observed between consecutive measurements.
3. Given that d changes by the same amount Δd, extract this value from recorded data.

To confirm if Figure 2 supports the prediction:

1. Analyze the plotted data points.
2. Draw a line of best fit through the data to determine the gradient.
3. If the gradient is negative and follows the form -k, the prediction is supported as it reflects the relationship R_c = k/d^2.
4. Explain any deviations or straightness in the line that could imply further experimental errors or accuracy in measurements.

The average count rate R_l is 100 s^-1, therefore:

1. Substitute the values into the formula:

   $t_d = \frac{R_l - R_c}{R_k R_c}$

2. Assuming a value for R_c equivalent to the previous calculations:

   $\text{Assume: } R_c = 0.76 s^{-1}$

   This may yield: (t_d = 0.01 \text{ s} \text{ (if R_k is 1)})

Radioactive decay is a random process where the emission of gamma photons can occur sporadically, not at consistent intervals. Even if 100 photons are emitted, some may not be detected due to the inherent dead time t_d, which might cause overlapping signals leading to missed detections. Hence, the assertion that all would be detected does not hold true under this random nature.