Exercise set

Radiation Dosimetry, Shielding, and Detector Exercises

Solved radiation exercises for absorbed dose, dose rate, inverse-square surveys, shielding, source decay, dead time, counting uncertainty and release gates.

Branch: Engineering Physics
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on radiation dosimetry, shielding and detector evidence: absorbed dose, dose rate, equivalent dose, inverse-square surveys, attenuation, source decay, detector dead time, background subtraction, counting uncertainty, contamination wipes, CTDI-style screening, weekly duty dose and guarded release gates. Plasma and charged-particle beam calculations are handled in a separate specialist exercise set.

Use the calculations as engineering screens only. Real radiation work requires approved procedures, calibrated instruments, legal dose limits, shielding authority, survey coverage, access control, uncertainty review and responsible sign-off.

How to use these exercises

Treat each exercise as a radiation-protection evidence check. Start by identifying the decision: dose estimate, area survey, shielding review, contamination release, detector correction, source inventory update, diagnostic QA screen, weekly occupancy review or access release. The same dose-rate number can have different meaning at a source surface, survey grid point, controlled-area boundary or occupied workstation.

For each problem, define:

source boundary: isotope, energy, activity date, geometry, duty factor and operating state;
detector boundary: calibration factor, energy response, background, count time, dead time and measurement range;
location boundary: distance, shielding path, scatter field, survey grid, contamination area and occupancy basis;
release boundary: accept, resurvey, extend count time, add shielding, restrict access, decontaminate, recalibrate or hold.

The goal is not to memorize dose formulas. The goal is to connect each result to the procedure, instrument and access decision that make radiation evidence defensible.

Release Evidence Notes

Radiation release evidence should state source, energy, activity date, geometry, survey location, detector calibration, background basis, count time, uncertainty, shielding condition, occupancy basis, contamination boundary, access state and interlock status. It should also identify whether the evidence supports routine operation, maintenance access, unrestricted release, restricted-area entry, imaging QA, material inspection, source storage or emergency response.

Preserve raw measurement context. A corrected detector reading without background, calibration date, energy range, detector settings, survey map and count-time evidence is weak. A passing average survey is also weak when the local maximum controls the release.

Engineering Boundary Notes

The exercises below simplify geometry, energy response, scatter, build-up and biological weighting. They do not replace radiation protection design, medical physics review, qualified surveys or regulatory compliance.

Inverse-square calculations assume a point-like source and weak scatter. Shielding calculations require material, thickness, energy spectrum, build-up, gaps, penetrations and geometry. Counting calculations assume suitable detector range, stable background and a rate low enough for the correction model. Contamination calculations depend on wipe efficiency, area definition and removable fraction.

If a result affects personnel access, patient or operator exposure, source release, room shielding or regulatory limits, treat the exercise result as a screen until the qualified procedure and calibrated survey evidence agree.

Common Release Mistakes

Common mistakes include using uncalibrated count rate as dose, ignoring background, applying inverse-square law in strong scatter fields, using nominal shielding without build-up or gaps, and releasing access without survey coverage and uncertainty margin.

Other release mistakes include:

using activity without correcting to the reference date;
applying a detector calibration factor outside its energy or geometry range;
accepting a shield calculation while ignoring penetrations, seams, doors, ducts or cable routes;
reporting a net count rate without count time, background basis and statistical uncertainty;
using a single clean grid point to release a whole area;
treating dead-time correction as reliable near the detector saturation region.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Dose and distance	1, 2, 3, 14, 15	Dose, dose rate, equivalent dose, CTDI and weekly duty	Check exposure magnitude and occupancy basis.
Shielding and source time	4, 5, 6, 7, 17	Attenuation, build-up, decay and uncertainty	Approve shield, update source activity or hold.
Detector and release evidence	8, 9, 10, 11, 12, 13, 16, 18	Dead time, background, counting precision, wipe and survey grid	Accept survey evidence or expand measurement.

Exercise 1: Absorbed Dose

An exposure deposits $3.2\times10^{-6}$ J in a 0.020 kg sample. Find absorbed dose.

Solution

D=\dfrac{E}{m}=\dfrac{3.2\times10^{-6}}{0.020}=1.60\times10^{-4}\ \text{Gy}=0.160\ \text{mGy}

Engineering Comment

Absorbed dose is energy per mass. It does not state dose distribution or biological weighting by itself.

Plausibility Check

Microjoules in tens of grams should give a small milligray-scale dose.

Exercise 2: Dose Rate

The dose in Exercise 1 occurs over 45 s. Find dose rate in mGy/h.

Solution

\dot{D}=0.160\dfrac{3600}{45}=12.8\ \text{mGy/h}

Engineering Comment

Dose rate is the relevant quantity for access time and occupancy decisions.

Plausibility Check

Forty-five seconds is 1/80 h, so the hourly rate is 80 times the dose.

Exercise 3: Equivalent Dose Screen

For photons, use radiation weighting factor $w_R=1$ . Find equivalent dose for 0.160 mGy.

Solution

H=w_R D=1(0.160)=0.160\ \text{mSv}

Engineering Comment

This screen changes for radiation types with different weighting factors.

Plausibility Check

For photons, gray and sievert numerical values match under this simplified screen.

Exercise 4: Inverse-Square Survey

Dose rate is 64 uSv/h at 1 m from a small source. Estimate dose rate at 4 m.

Solution

\dot{D}_2=64\left(\dfrac{1}{4}\right)^2=4.0\ \mu\text{Sv/h}

Engineering Comment

The inverse-square law is weak near extended sources, walls or strong scatter.

Plausibility Check

Four times the distance gives one sixteenth of the rate.

Exercise 5: Simple Shield Attenuation

A shield has linear attenuation coefficient 0.42 cm^-1 and thickness 5 cm. Ignore build-up. Find transmission.

Solution

T=e^{-\mu x}=e^{-0.42(5)}=0.122

Engineering Comment

Uncollided attenuation can be non-conservative when scattered photons matter.

Plausibility Check

Two attenuation lengths gives transmission near 0.1.

Exercise 6: Build-Up Corrected Dose Rate

Incident dose rate is 80 uSv/h. Transmission is 0.122 and build-up factor is 1.4. Find transmitted rate.

Solution

\dot{D}=80(0.122)(1.4)=13.7\ \mu\text{Sv/h}

Engineering Comment

Build-up reminds the reviewer that shielding is not only exponential absorption.

Plausibility Check

The result is higher than uncollided 9.8 uSv/h but lower than incident rate.

Exercise 7: Source Decay

A source has activity 120 MBq and half-life 30 days. Find activity after 45 days.

Solution

A=120\left(\dfrac{1}{2}\right)^{45/30}=42.4\ \text{MBq}

Engineering Comment

Activity records should include reference date and isotope half-life.

Plausibility Check

After 1.5 half-lives, activity should be between one quarter and one half of the original.

Exercise 8: Activity Date Uncertainty

If the same source date is uncertain by 3 days, estimate fractional activity uncertainty using $\lambda=\ln 2/30$ day^-1.

Solution

\dfrac{\Delta A}{A}\approx \lambda\Delta t=\dfrac{\ln2}{30}(3)=0.069=6.9\%

Engineering Comment

Short-lived sources need tightly controlled reference dates.

Plausibility Check

Three days is one tenth of a half-life, so several percent uncertainty is plausible.

Exercise 9: Detector Dead-Time Correction

A detector records 18,000 cps with nonparalyzable dead time $6.0\times10^{-6}$ s. Find corrected rate.

Solution

R=\dfrac{R_m}{1-R_m\tau}=\dfrac{18000}{1-18000(6.0\times10^{-6})}=20179\ \text{cps}

Engineering Comment

Dead-time correction becomes sensitive as measured rate approaches the detector limit.

Plausibility Check

The corrected rate must be higher than the measured rate.

Exercise 10: Background Subtraction

Gross count is 5200 counts in 100 s. Background is 700 counts in 100 s. Find net count rate.

Solution

R_n=\dfrac{5200-700}{100}=45\ \text{cps}

Engineering Comment

Background basis should match location, time and detector settings.

Plausibility Check

Net counts are 4500 over 100 s.

Exercise 11: Counting Uncertainty

Using gross 5200 and background 700 counts, estimate standard uncertainty in net counts.

Solution

\sigma_N=\sqrt{5200+700}=76.8\ \text{counts}

Engineering Comment

Poisson counting uncertainty matters near release limits.

Plausibility Check

The uncertainty is near the square root of total counted events.

Exercise 12: Required Counting Time

At net rate 45 cps, estimate time to collect 10,000 net counts.

Solution

t=\dfrac{10000}{45}=222\ \text{s}

Engineering Comment

Longer count time can reduce relative statistical uncertainty but not calibration bias.

Plausibility Check

At about 45 counts each second, a few minutes are needed.

Exercise 13: Survey Grid Release

A grid release limit is 0.50 uSv/h above background. Four net readings are 0.22, 0.31, 0.48 and 0.56 uSv/h. Decide.

Solution

\max=0.56\ \mu\text{Sv/h}>0.50\ \mu\text{Sv/h}

The grid fails.

Engineering Comment

Release is controlled by the local maximum, not only the average.

Plausibility Check

One reading is above the limit, so the decision is hold.

Exercise 14: Wipe-Test Contamination

A wipe has 240 net counts in 120 s. Detector efficiency is 0.30 and wiped area is 100 cm2. Estimate activity density.

Solution

A_s=\dfrac{240/120}{0.30(100)}=0.0667\ \text{Bq/cm}^2

Engineering Comment

Wipe efficiency and removable fraction should be stated in the procedure.

Plausibility Check

Two cps divided by efficiency gives 6.67 Bq over 100 cm2.

Exercise 15: CTDI-Style Dose Screen

Four peripheral readings are 18, 20, 19 and 21 mGy, and center reading is 12 mGy. Compute weighted CTDI screen $(center+2\overline{periphery})/3$ .

Solution

\overline{P}=\dfrac{18+20+19+21}{4}=19.5\ \text{mGy}

CTDI_w=\dfrac{12+2(19.5)}{3}=17.0\ \text{mGy}

Engineering Comment

This is a QA screen, not a patient-specific dose estimate.

Plausibility Check

The weighted result lies between center and peripheral values.

Exercise 16: Pulsed Weekly Dose

A pulsed x-ray system gives 2.0 mSv/h during pulses. Duty factor is 0.015 and occupied time is 20 h/week. Estimate weekly dose.

Solution

H_w=2.0(0.015)(20)=0.60\ \text{mSv/week}

Engineering Comment

Duty factor must come from measured or controlled pulse timing, not intent alone.

Plausibility Check

The average rate is 0.03 mSv/h, so 20 h gives 0.60 mSv.

Exercise 17: Calibration Factor

A detector reads 92 uSv/h in a reference field of 100 uSv/h. Find calibration factor to multiply readings.

Solution

k=\dfrac{100}{92}=1.087

Engineering Comment

Calibration factor should be energy- and geometry-appropriate.

Plausibility Check

The detector under-reads, so the correction factor is above 1.

Exercise 18: Radiation Release Gate

A release requires all grid readings below limit, dead-time fraction below 10 percent and calibration current. Results are one grid fail, dead-time fraction 8 percent and calibration current. Decide.

Solution

\text{grid}=\text{fail},\qquad 8\%<10\%,\qquad \text{calibration}=\text{current}

The release fails because the grid survey fails.

Engineering Comment

Good detector status cannot compensate for a failed area survey.

Plausibility Check

One required criterion fails, so the release must be held.

Validation Package Checklist

source activity, date, isotope and geometry are recorded;
detector calibration, background and count time are stated;
shielding model includes thickness, material and uncertainty limits;
survey coverage identifies local maxima, not only averages;
contamination measurements state area and efficiency assumptions;
dead-time, energy response, detector range and calibration factor are appropriate for the field being measured;
shielding evidence includes scatter, build-up, penetrations, gaps, occupancy and access state when relevant;
counting uncertainty, background variability and guard bands are included near release limits;
contamination evidence states wipe method, removable fraction assumption and decontamination response;
release decision states accept, resurvey, extend count time, add shielding, restrict access, decontaminate, recalibrate or hold.

The final release statement should name the controlling measurement. Good detector calibration cannot compensate for a failed survey grid, and a nominal shielding calculation cannot release an area when measured local maxima exceed the limit.

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