Interactive practical

Inverse-square law for gamma radiation

Measure background and gamma counts at fixed source-detector distances, then test corrected count rate against inverse-square distance.

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A-Level Physics

Inverse-square law for gamma radiation

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Practical focus

Measure corrected gamma count rate at different distances and test an inverse-square relationship.

Practical method

Aim

Investigate how the count rate from a gamma source varies with distance and test the inverse-square law.

Variables

Independent variable: distance d from source to GM tube
Dependent variable: corrected count rate C
Control variables: the same source; counting time; background subtracted.

Apparatus

Sealed gamma source (cobalt-60) in a holder
Geiger–Müller tube and counter
Metre rule
Tongs
Stopwatch

Method

Measure the background count rate with no source present (several minutes).
Your teacher places the source in a holder facing the GM tube using tongs.
Measure the count rate at distances from 0.10 m to 0.30 m, counting for long enough to record several hundred counts.
Subtract background from each reading.
Plot \(\dfrac{1}{\sqrt{C}}\) against d: a straight line whose intercept on the d axis gives the unknown offset e between the marked and actual positions of the source.

Sample results

Background 0.50 counts s⁻¹ already subtracted.

d / m	C / counts s⁻¹	1/√C / s\(^{1/2}\)
0.10	200.0	0.0707
0.15	91.8	0.1044
0.20	52.5	0.1380
0.25	33.9	0.1717
0.30	23.7	0.2054

Analysis

\(\dfrac{1}{\sqrt{C}}\) against d is a straight line, so C ∝ \(\dfrac{1}{(d + e)^{2}}\): the inverse-square law. The line meets the d axis at −0.005 m, so the source is 5 mm further inside its holder than the marked position.

Uncertainty

Radioactive decay is random: the uncertainty in N counts is about √N. Counting 2000 counts gives about 2%; the furthest point needs a longer count. The offset e is a systematic error removed by the graph.

Errors and improvements

Problem	Effect	Improvement
Random nature of decay	Scatter	Count for longer; repeat
Background radiation	Count rate too high at large d	Measure and subtract background
Unknown position of the source inside its holder	Systematic offset	Plot \(\dfrac{1}{\sqrt{C}}\) against d and find e from the intercept

Risk assessment

Hazard	Risk	Control measure
Ionising radiation	Cell damage	Use tongs, hold the source at arm's length, minimise time, return it to its lead box promptly
Contamination	Ingestion	Sealed sources only; wash hands

Common mistakes

Forgetting to subtract background.
Plotting C against d and calling it inverse square.
Counting for too short a time at large distances.

Worked analysis

Gradient of \(\dfrac{1}{\sqrt{C}}\) against d = \(\dfrac{0.2054 - 0.0707}{0.20}\) = 0.674. Intercept on the d axis: 0.10 − \(\dfrac{0.0707}{0.674}\) = −0.005 m, so e = 5 mm.

Common misconception

The inverse-square law applies to gamma rays from a point source because they spread out in all directions and are hardly absorbed by air; it does not apply to alpha or beta.

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