Inverse Square Law

The inverse square law states that radiation intensity from an ideal point source decreases in proportion to the square of the distance from the source. As distance increases, radiation spreads over a larger area, so less radiation passes through each unit area. Formula: I ∝ 1/d².

Radiation spreading uniformly from a point source covers the surface of an expanding sphere. The surface area of a sphere is A = 4πr². Doubling the distance increases the area fourfold, so the same emitted energy is distributed over four times the area and intensity falls to one-quarter.

The calculation formula is I1/I2 = d2²/d1², or I2 = I1 × (d1/d2)². I1 is the original intensity, I2 is the new intensity, d1 is the original distance and d2 is the new distance. Both distances must be measured from the same source reference point.

I represents radiation intensity, d represents distance, and subscripts 1 and 2 distinguish the initial and final conditions. The square symbol (²) means multiply the distance by itself. The law can also be applied to dose rate when its assumptions are satisfied.

If the distance from a point source doubles, intensity decreases to one-quarter of its original value. Example: increasing distance from 1 m to 2 m gives I2 = I1 × (1/2)² = I1/4. If the initial dose rate is 80 µSv/h, the new dose rate is 20 µSv/h.

If the distance triples, intensity decreases to one-ninth of its original value. Example: increasing distance from 1 m to 3 m gives I2 = I1 × (1/3)² = I1/9. An initial intensity of 90 units becomes 10 units.

If distance is halved, intensity increases by a factor of four. Example: moving from 2 m to 1 m gives I2 = I1 × (2/1)² = 4I1. An initial dose rate of 15 µSv/h becomes 60 µSv/h under ideal point-source conditions.

Distance and intensity have an inverse-square relationship, not a simple inverse relationship. Doubling distance does not halve intensity; it reduces it to one-quarter. Increasing distance by a factor of four reduces intensity to one-sixteenth.

First, identify the initial intensity or dose rate and both distances. Second, write I2 = I1 × (d1/d2)². Third, substitute the numbers using consistent distance units. Fourth, square the distance ratio and multiply by the original intensity. Finally, check whether the result should increase or decrease.

A dose rate is 120 µSv/h at 1 m. Calculate the dose rate at 3 m. Formula: I2 = 120 × (1/3)² = 120 × 1/9 = 13.3 µSv/h. The answer is lower because the distance increased.

A dose rate is 5 mGy/h at 4 m. Calculate the dose rate at 2 m. Formula: I2 = 5 × (4/2)² = 5 × 4 = 20 mGy/h. The dose rate increases because the distance has been halved.

Radiation intensity is 200 units at 50 cm. Find the intensity at 100 cm. Formula: I2 = 200 × (50/100)² = 200 × 0.25 = 50 units. Doubling the distance reduces intensity to one-quarter.

The inverse square law can be rearranged to calculate distance. Formula: d2 = d1 × √(I1/I2). Example: if intensity must decrease from 100 units at 1 m to 25 units, d2 = 1 × √(100/25) = 2 m.

A point source produces a dose rate of 64 µSv/h at 1 m. At what distance would the dose rate be 4 µSv/h? d2 = 1 × √(64/4) = √16 = 4 m. Increasing distance to 4 m reduces the dose rate by a factor of 16.

Dose rate describes radiation dose received per unit time, such as mGy/h or µSv/h. For a point source in free space, dose rate approximately follows the inverse square law. Increasing distance reduces the dose received during the same exposure time.

Total dose equals dose rate multiplied by exposure time when the dose rate is constant. Formula: D = dose rate × time. Example: 20 µSv/h for 0.5 hours gives 10 µSv. Reducing exposure time reduces accumulated dose, while increasing distance can reduce dose rate.

Distance is one of the three fundamental radiation protection principles: time, distance and shielding. Increasing distance from a radiation source can substantially reduce exposure. This is particularly useful around small radioactive sources where inverse-square behaviour is a reasonable approximation.

In external beam radiotherapy, source-to-surface distance (SSD) and source-to-axis distance (SAD) are important geometric measurements. Changes in distance influence photon fluence and beam intensity. Clinical dose calculations also account for attenuation, scatter, beam geometry and treatment-machine characteristics.

SSD is the distance from the radiation source to the patient's surface along the beam axis. Changing SSD affects the beam geometry and the incident intensity. In actual treatment planning, dose changes cannot always be calculated using the simple inverse square law alone.

SAD is the distance from the radiation source to the machine isocentre, the point around which the gantry rotates. Isocentric treatments use this reference geometry. Understanding SAD helps explain beam divergence and distance-dependent intensity changes.

The simple inverse square law assumes a point source, geometric spreading and no significant absorption or scattering between source and measurement point. Real radiation fields may involve extended sources, collimation, attenuation and scatter. Therefore, the law is an approximation in some clinical situations.

Remember that increasing distance decreases intensity and decreasing distance increases intensity. Always square the distance ratio, keep units consistent and check whether your answer is physically sensible. Common mistakes include using the ratio upside down, forgetting to square it and confusing dose with dose rate.

I(2) = I(1) (d(1) / d(2)) squared