To find when a detector reaches a target count rate, subtract the background from both the initial reading and the target reading. Apply the decay law to the remaining source count rates.
This model treats the background count rate as constant and the detector’s corrected count rate as proportional to the source activity. The observed reading approaches the background as the source contribution falls.
- Write the initial, target and background rates in the same units.
- Subtract the background from both readings.
- Use the half-life and a logarithm when the ratio is not an exact power of two.
Worked example | Find the time to a reading of 50 counts per second
The observed count rate starts at 180 counts/s, the background is 20 counts/s and the source half-life is 6 minutes. The target reading is 50 counts/s. The initial source rate is 180 − 20 = 160 counts/s and the target source rate is 50 − 20 = 30 counts/s.
Solve 30 = 160 × (½)t/6. Taking logarithms gives t = 6 ln(30/160) ÷ ln(½) ≈ 14.5 minutes.
Bracket the answer with two whole half-lives
After 12 minutes, the source rate is 160 ÷ 4 = 40 counts/s, and the observed rate is 60 counts/s. After 18 minutes, the source rate is 20 counts/s, and the observed rate is 40 counts/s. A target of 50 counts/s falls between these two readings.
Try a target with an exact number of half-lives
The initial observed rate is 108 counts/s. The background rate is 12 counts/s and the half-life is 4 minutes. When does the observed rate reach 36 counts/s?
Answer. The source rate goes from 96 to 24 counts/s, a reduction to one quarter. That takes two half-lives, or 8 minutes.
Frequently asked questions
What if the target equals the background?
The ideal exponential source contribution tends to zero without reaching zero in finite time. The modelled observed rate tends to the background.
Why use the term modelled reading?
Actual counts fluctuate. This calculation predicts when the expected count rate reaches the target, using a constant background.
Official information
SEAB, 2027 GCE A-Level H2 Physics syllabus (9478), Nuclear Physics, radioactive decay and half-life. Syllabus checked 11 November 2026. The examples are original.