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

To determine the corrected count rate Rc, we use the formula:
[ R_c = R - R_b ]
where R is the total count rate and R_b is the background count rate.

First, calculate the uncorrected count rate for d = 530 mm:
[ R = \frac{C_1 + C_2 + C_3}{3} = \frac{90 + 117 + 102}{3} = 103 \text{ counts in } 100 s \approx 1.03 \text{ counts/s} ]
The background count C1 needs to be converted to a rate:
[ R_b = \frac{630}{15 \times 60} \approx 0.7 \text{ counts/s} ]
Thus, if we compute:
[ R_c = 1.03 - 0.7 = 0.33 \text{ s}^{-1} ]
This confirms that Rc is approximately 0.33 s⁻¹.

The student's prediction states that Rc is inversely proportional to the square of the distance d. We can analyze the counts for the two distances:

• At d = 530 mm, Rc is calculated to be approximately 0.33 s⁻¹.
• At d = 380 mm, Rc shows a value of 0.76 s⁻¹.

Using the prediction formula ( R_c = \frac{k}{d^2} ):
[ RC \propto \frac{1}{d^2} ]
will imply that as d decreases, Rc should increase.
Since Rc increased from approximately 0.33 s⁻¹ to 0.76 s⁻¹ when d changed from 530 mm to 380 mm, the values do support the student's prediction.

1. To safely reduce d, the student should adjust the position of the detector downwards, ensuring it is securely clamped at each measurement to prevent falls.

2. It is essential to lower slowly and take care to have all connections checked to avoid any accidental movement of the sealed source while making adjustments.
   This approach minimizes the risk of radiation exposure while ensuring accurate measurements.

Δd can be determined from the data points indicated in Figure 2. If the values of d have been plotted and the changes noted after each measurement, then simply take the difference between two successive values of d.
For instance, if d changes from 380 mm to 440 mm, then:
[ Δd = 440 ext{ mm} - 380 ext{ mm} = 60 ext{ mm} ] Assuming consistent intervals, the value of Δd would be consistent across measurements.

To confirm the prediction illustrated by Figure 2, the student can perform the following steps:

1. Confirm that the plotted graph shows a linear relationship between ( \log(R_c) ) and ( \log(d) ).
2. Calculate the slope (gradient) of the line; it should equate to -2 if the prediction is correct, reflecting the inverse square relationship.
3. If the gradient does not equal -2, it would imply that the relationship does not support the student's prediction.

Using the provided information, the average measured count rate Rₗ is 100 s⁻¹.
The average photon rate that enters the detector is 2 per second.
To calculate the dead time ( tₗ ):
[ t_𝑙 = \frac{R_l - R_c}{R_c R_2} ]
Assuming that R₂ is the corrected measure, we substitute the known values, leading to a calculation of ( t_𝑙 ).
In this case, the specific value will require the measured corrections to compute.

The student’s assertion relies on the assumption that all incoming photons would be detected without interruption. However, due to the random nature of radioactive decay, even if 100 photons are emitted in one second, there is a statistical likelihood that not all photons will arrive at the detector within that timeframe.
The dead time associated with the detector means that some emitted photons could arrive within the time window where the detector is still busy processing previously detected photons.
Therefore, some gamma photons are inevitably lost, leading to detection rates that do not equal the emission rates.