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The inverse square law and radiology basics, worked out

Study cards with fictional worked examples: the inverse square law, X-ray photon energy and tube voltage, gray, sievert, rad and rem, activity and half-life, ALARA, and how CT differs from MRI. Then practice questions with explanations and a summary that tells you which cards to reread. Textbook physics for study only. Free, no sign-up needed.

  • 6 study cards
  • Worked examples
  • 20 practice questions
  • No sign-up needed
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1 · Study cards

The ideas and formulas, with worked examples

6 cards, each with the idea in plain words and fictional worked examples. Every example is worked out step by step. Jump to the card you need:

Study card 1

The inverse square law

A small source sends its energy out in every direction. Farther away, the same energy is spread over the surface of a bigger and bigger sphere, so each square meter receives less. That is why intensity from a point source falls with the square of the distance.

The key ideas

  1. Intensity

    Power per unit area: I = P ÷ A, in watts per square meter (W/m²).

    Example: 10 W spread over 2 m² is 5 W/m².

  2. Point source

    Spreading evenly over a sphere of area 4πr²: I = P ÷ (4πr²). So I is proportional to 1 ÷ r².

    Example: Double r and the sphere has 4 times the area, so I is one quarter.

  3. Two distances, one source

    I₂ = I₁ × (r₁ ÷ r₂)². Use the same units for both distances.

    Example: 36 at 1 m becomes 36 × (1 ÷ 3)² = 4 at 3 m.

  4. Finding a distance

    r₂ = r₁ × √(I₁ ÷ I₂). A weaker reading means a larger distance.

    Example: 200 at 1 m and 50 at r₂: r₂ = 1 × √4 = 2 m.

Worked examples

  1. A small lamp-like source gives an intensity of 36 W/m² at 1 m. What is the intensity at 3 m?

    1. Distance is 3 times larger, so intensity is 1 ÷ 3² = 1 ÷ 9 as large.
    2. 36 ÷ 9 = 4.

    Answer: 4 W/m²

  2. The intensity is 80 W/m² at 2 m from a small source. What is it at 5 m?

    1. I₂ = 80 × (2 ÷ 5)² = 80 × 0.16.
    2. 80 × 0.16 = 12.8.

    Answer: 12.8 W/m²

  3. A reading drops from 200 units at 1 m to 50 units at a farther point. How far is that point?

    1. The intensity fell by a factor of 200 ÷ 50 = 4.
    2. The distance grew by √4 = 2, so the point is 2 × 1 = 2 m away.

    Answer: 2 m

  4. An ideal point source radiates 100 W evenly in all directions. What is the intensity 2 m away?

    1. Sphere area = 4π × 2² = 16π ≈ 50.27 m².
    2. I = 100 ÷ 50.27 ≈ 1.99.

    Answer: About 1.99 W/m²

Watch out: The law fits a small source spreading freely in all directions, with nothing absorbing or scattering the energy on the way. Absorption or shielding in the way changes the numbers, so it is the starting model. Also square the ratio: tripling the distance divides by 9, not 3.

Study card 2

Photon energy and how an X-ray tube sets it

Light and X-rays are electromagnetic waves that also arrive as packets called photons. Shorter wavelength means more energy per photon. In an X-ray tube, electrons accelerated through a high voltage strike a metal anode and some of their energy becomes X-ray photons.

The key ideas

  1. Photon energy

    E = hf = hc ÷ λ. Shorter wavelength, higher energy.

    Example: Halve the wavelength and the energy doubles.

  2. The shortcut

    hc = 1240 eV·nm, so E (in eV) = 1240 ÷ λ (in nm).

    Example: λ = 0.1 nm gives 1240 ÷ 0.1 = 12,400 eV = 12.4 keV.

  3. Tube voltage and maximum photon energy

    Electrons accelerated through V volts carry at most qV of energy, so the most energetic photon is qV: a 50 kV tube gives photons up to 50 keV.

    Example: 80 kV means at most 80 keV.

  4. Shortest wavelength from a tube

    λ(min) = 1240 ÷ (maximum energy in eV), in nm.

    Example: 62 kV: 1240 ÷ 62,000 = 0.02 nm.

  5. Ionizing radiation

    X-rays can knock electrons out of atoms, so they are called ionizing. Radio waves and visible light are non-ionizing.

    Example: A single X-ray photon carries far more energy than a visible-light photon.

Worked examples

  1. An X-ray photon has a wavelength of 0.062 nm. What is its energy?

    1. E = 1240 ÷ 0.062 eV.
    2. 1240 ÷ 0.062 = 20,000 eV = 20 keV.

    Answer: 20 keV

  2. A tube runs at 50 kV. What is the largest photon energy it can produce?

    1. Maximum photon energy = qV, which is 50 keV for 50 kV.

    Answer: 50 keV

  3. For that 50 kV tube, what is the shortest X-ray wavelength?

    1. E(max) = 50,000 eV.
    2. λ(min) = 1240 ÷ 50,000 = 0.0248 nm.

    Answer: 0.0248 nm

Watch out: Keep the units straight: 1240 eV·nm needs eV and nm together. A tube voltage in kV sets the largest photon energy in keV.

Study card 3

Radiation units: gray, sievert, rad and rem

Absorbed dose counts energy deposited per kilogram. Dose equivalent adjusts it by how much biological effect a kind of radiation has (the RBE). Each has an SI unit and an older unit, and the conversions are a hundred.

The key ideas

  1. Gray (Gy)

    SI unit of absorbed dose: 1 Gy = 1 J/kg.

    Example: 0.2 J absorbed in 4 kg is 0.05 Gy.

  2. Rad

    Older unit of absorbed dose: 1 Gy = 100 rad.

    Example: 0.05 Gy = 5 rad.

  3. Sievert (Sv)

    SI unit of dose equivalent: Sv = Gy × RBE.

    Example: For X-rays the RBE is 1, so 0.4 Gy is 0.4 Sv.

  4. Rem

    Older unit of dose equivalent: 1 Sv = 100 rem, and rem = rad × RBE.

    Example: 0.3 Sv = 30 rem.

Worked examples

  1. A block of tissue-like material with a mass of 4 kg absorbs 0.2 J from a beam. What is the absorbed dose in gray and in rad?

    1. Dose = energy ÷ mass = 0.2 ÷ 4 = 0.05 Gy.
    2. 0.05 Gy × 100 = 5 rad.

    Answer: 0.05 Gy, which is 5 rad

  2. An absorbed dose of 0.02 Gy comes from X-rays. What is the dose equivalent in sievert and in rem?

    1. The RBE for X-rays is 1, so 0.02 Gy × 1 = 0.02 Sv.
    2. 0.02 Sv × 100 = 2 rem.

    Answer: 0.02 Sv, which is 2 rem

Watch out: Gray and sievert are not interchangeable: the sievert includes the RBE. For X-rays and gamma rays the RBE is 1, so the numbers match, but that is not true of every type of radiation.

Study card 4

Activity and half-life

Activity says how many nuclei in a sample decay each second. Half-life is the time for half of the radioactive nuclei in a sample to decay, so the amount (and the activity) halves again every half-life.

The key ideas

  1. Becquerel (Bq)

    SI unit of activity: 1 Bq = 1 decay per second.

    Example: 640 Bq means 640 decays each second.

  2. Curie (Ci)

    Older unit: 1 Ci = 3.70 × 10¹⁰ Bq.

    Example: 7.4 × 10¹⁰ Bq is 2 Ci.

  3. Half-life

    The time for half of the nuclei to decay. After n whole half-lives, the fraction left is (1 ÷ 2)ⁿ.

    Example: After 3 half-lives: 1 ÷ 8 remains.

Worked examples

  1. A sample has 800 units of a radioactive material. How much is left after 3 half-lives?

    1. Halve three times: 800 → 400 → 200 → 100.

    Answer: 100 units

  2. A source has an activity of 640 Bq and a half-life of 6 hours. What is its activity 18 hours later?

    1. 18 ÷ 6 = 3 half-lives.
    2. 640 → 320 → 160 → 80.

    Answer: 80 Bq

Watch out: Half-life is not half of the time to disappear: after two half-lives a quarter is left, not nothing. The whole-number rule above works for whole half-lives.

Study card 5

ALARA: time, distance and shielding

ALARA stands for "as low as reasonably achievable." Radiation-safety references describe three basic factors: time near a source, distance from it, and shielding between you and it. This card is textbook background and uses made-up units, not real doses.

The key ideas

  1. ALARA

    A risk-management philosophy: avoid radiation exposure that gives no direct benefit, even at small amounts.

    Example: The letters stand for "as low as reasonably achievable."

  2. Time

    Total exposure = exposure rate × time. Less time means less total.

    Example: 5 units per hour for 3 hours = 15 units.

  3. Distance

    The rate falls with the square of the distance, so doubling the distance cuts the rate to a quarter.

    Example: 40 units per hour at 1 m is 10 at 2 m.

  4. Shielding

    A barrier between source and person. The right material depends on the type of radiation.

    Example: Alpha particles cannot get through the outer layer of skin; gamma rays can need several inches of dense material such as lead.

Worked examples

  1. In a textbook problem, a source gives a rate of 40 units per hour at 1 m. What total is received over 3 hours at 2 m?

    1. At 2 m the rate is 40 ÷ 2² = 10 units per hour.
    2. Total = rate × time = 10 × 3 = 30 units.

    Answer: 30 units

Watch out: This page is for study. It does not tell anyone how to handle a real source or what exposure is acceptable; those decisions follow a workplace’s own radiation-safety program and training.

Study card 6

CT versus MRI

Both make cross-section pictures of the body, but they work in completely different ways. CT uses X-rays, which are ionizing. MRI uses a strong magnetic field and radio waves.

The key ideas

  1. CT (computed tomography)

    An X-ray source rotates around the body while digital detectors opposite it record the beams; a computer reconstructs flat slices that can be stacked into a 3-D image.

    Example: Uses ionizing radiation (X-rays).

  2. MRI (magnetic resonance imaging)

    Powerful magnets align protons in the body; radio-frequency pulses stimulate them, and sensors detect the energy released as they realign.

    Example: Does not emit the ionizing radiation found in X-ray and CT imaging, but does use a strong magnetic field.

  3. Ionizing vs non-ionizing

    Ionizing radiation has enough energy to knock electrons out of atoms. Non-ionizing radiation (radio waves, visible light, microwaves) does not.

    Example: X-rays: ionizing. Radio waves: non-ionizing.

Worked examples

  1. Which of CT and MRI relies on X-rays, and which on radio waves?

    1. CT builds slices from X-ray beams recorded by detectors.
    2. MRI uses radio-frequency pulses inside a strong magnetic field.

    Answer: CT: X-rays. MRI: magnetic field and radio waves.

Watch out: Do not mix up the energy sources: MRI is not "a different kind of X-ray." It is a different physical method. This card says nothing about which scan suits which situation; that is a clinical decision.

2 · Practice questions

20 practice questions

Questions across the cards, in card order. Most are numbers to work out by hand and type (no calculator needed); the rest are multiple choice. Each answer is explained at once, and the end is a summary by topic that says which cards to reread. Your result is free: no sign-up needed.

Question 1 of 20 · Inverse square law

A small source gives an intensity of 36 units at 1 m. What is the intensity at 3 m?

Number (units are the same as in the question)

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Sources

Where the facts come from

The physics and units on this page follow these open textbooks and public health pages, in our own words. The scenarios, numbers and worked examples are our own and invented. This is study material for radiography and physics courses: it gives no patient-dose figures and no advice about real exposure, so follow your own course materials and training for anything practical.

Rusty on the arithmetic behind squares, roots and ratios? Our free math refresher covers it step by step. Heading back to school and not sure which program fits? The free skills finder turns what you have already done into a skills list.

Questions

Common questions

For a small source that spreads its energy freely in all directions, intensity is proportional to 1 ÷ r², where r is the distance from the source. The energy is spread over the surface of a sphere of area 4πr², so double the distance and the intensity is one quarter. Written for two distances: I₂ = I₁ × (r₁ ÷ r₂)². For example, 36 at 1 m becomes 4 at 3 m.

Square the ratio of the distances. Intensity at the new distance = old intensity × (old distance ÷ new distance)². To find a distance from two readings, r₂ = r₁ × √(I₁ ÷ I₂). With 200 at 1 m and 50 at the farther point, the intensity fell by 4, so the distance grew by √4 = 2: the point is 2 m away. Use the same distance unit throughout.

Gray (Gy) measures absorbed dose: 1 Gy = 1 joule of energy per kilogram. Sievert (Sv) measures dose equivalent, which multiplies the gray by the RBE of the type of radiation: Sv = Gy × RBE. The textbook lists an RBE of 1 for X-rays and gamma rays, so the numbers match for them. The older units are rad (100 rad = 1 Gy) and rem (100 rem = 1 Sv).

Activity, the number of decays per second in a radioactive sample. 1 becquerel (Bq) is 1 decay per second. The curie is the older unit: 1 Ci = 3.70 × 10¹⁰ Bq.

Half-life is the time for half of the radioactive nuclei in a sample to decay. After one half-life half remain, after two a quarter, after three an eighth: the fraction left is (1 ÷ 2)ⁿ for n whole half-lives. A source of 640 Bq with a 6-hour half-life is at 80 Bq after 18 hours.

In an X-ray tube, electrons are accelerated through a high voltage and strike an anode, and some of their energy becomes X-ray photons. The maximum photon energy equals the electron energy, qV: a 50 kV tube gives photons up to 50 keV. Photon energy and wavelength are linked by E = hc ÷ λ with hc = 1240 eV·nm, so 50 keV corresponds to a shortest wavelength of about 0.0248 nm.

ALARA stands for "as low as reasonably achievable." The CDC describes it as the risk-management philosophy of radiation safety, and names three basic factors: time near a source, distance from it and shielding. The distance factor is the inverse square law at work. This page is study material and does not give advice about any real exposure.

CT rotates an X-ray source around the body, records the beams on digital detectors and reconstructs slices on a computer, so it uses ionizing radiation. MRI uses a strong magnetic field and radio waves and does not emit the ionizing radiation found in X-ray and CT imaging. This page explains the physics only; which scan is used for what is a clinical decision.

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