Inverse Square Law Simulator

What Is the Inverse Square Law Simulator?

How to Use the Simulator

1. Read the four cards before you touch anything. They are already answering for a 100 W point source one metre away: Intensity at the detector 7.958 W/m^2, Against the reading at 1 m 1.0000x, Doubling the distance leaves 1/4 exactly and If you measured from the surface 1.0000x.
2. Read the sentence under each card, because that is where the panel shows its hand. They read P / (4 pi r^2) -- the surface the energy crosses is a sphere, area 4 pi r^2, the reading at this distance against the reading at 1 m, double the distance and a point source leaves 1/4 exactly and no error: the source is a point, so r from the centre and r from the surface are the same.
3. Move Distance r, from the centre one rung right, from 1 m to 2 m. The first card falls to 1.989 W/m^2 and the second to 0.2500x. Watch the sphere row under the drawing while you do it: it goes from 12.566 m^2 here to 50.265 m^2 here, which is four times for a reading that fell to a quarter.
4. Put the distance back and read the six grid rows. On the opening state they are 2 for the exponent, a sphere, area 4 pi r^2, 12.566 m^2 here, 1.414 m for the halving, 70.711 cm for the doubling, 2.821 m for where it would read 1 W/m^2, and 12.566 W for the power that would give 1 W/m^2 here. Those last two are not the same kind of number as each other.
5. Move Shape of the source one rung. The slider label turns to line source over exponent 1 -- the surface is a cylinder, area 2 pi r L, the cost card turns to 1/2 exactly, the surface row to a cylinder, area 2 pi r L, growing as r^1, and the halving distance stretches from 1.414 m to 2.000 m.
6. Push it one further, to the flat sheet. The cost card reads 1x -- no change, and the three distance rows stop being numbers at all: no distance halves it, no distance doubles it and every distance reads the same. They are sentences from branches of their own, never a division.
7. Push it to the top rung, the radar echo. The exponent row reads 4, the cost card 1/16 exactly — not an eighth — and the surface row a sphere twice over, out and back, growing as r^2, because the echo pays for the sphere on the way out and again on the way back.
8. Notice what did not change while you did all that. At one metre the first card reads 7.958 W/m^2 on every one of the four shapes. That is the anchoring, and the sentence under the card admits it: on the three non-point rungs it begins anchored to the same reading at 1 m instead of quoting the law.
9. Now walk Source size, as R / s from end to end. The label goes from a true point source to source radius 1.0000 of the distance, the fourth card from 1.0000x to 4.0000x, and nothing else on the panel moves at all. The bottom band redraws, because that card is the only reading in the lab that depends on this slider.
10. Finally push Source power P to each end. At its bottom rung the first card reads 79.577 uW/m^2 and at its top 7.958 kW/m^2, while the ratio card sits at 1.0000x and the halving row at 1.414 m throughout. Power scales the reading and decides nothing about how it falls off.
11. Press Reset when you have finished. It puts the four sliders back to 100.000 W, 1.000 m, point source and a true point source, and replays the opening settle so the scene is drawn in rather than cut to.

Dragging is the right way to find out which readings move together and a poor way to land on a particular state, because all four controls step by whole rungs. The five preset buttons above exist for that reason, and the written inverse square law guide is where the argument is made from scratch and the problems are worked with every step shown.

One thing this panel cannot do is take a power, a distance or a source size that is not on a rung, and it cannot run the relation backwards at all. The inverse square law calculator takes typed figures and solves in four directions, including the one question this lab never asks: at what distance has a known reading fallen to a target.

Worked Example: One Metre, Then Two

Nothing in the tables below was worked out here. Each column is a state the lab was put into through its own sliders and then read back from, cell by cell, so every string in them is the panel’s own. The first table is the pair this section is named after.

One: one source, one geometry, and one rung of the distance ladder

Press A 100 W point source one metre away and then Twice as far: exactly a quarter. Only the distance slider moves between them, so the four control rows are the same except one. This table is transposed, because the comparison worth making runs down a column.

The sphere row is half the lesson and leaving it out would hide the reason. The reading falls from 7.958 W/m^2 to 1.989 W/m^2, which is exactly a quarter, and the surface the energy crosses goes from 12.566 m^2 here to 50.265 m^2 here, which is exactly four times. The same power, four times the area, a quarter of the reading: that is the whole law in two rows of one table.

Two rows refuse to move, and they are the interesting ones. Doubling the distance leaves reads 1/4 exactly in both columns, because it is a property of the geometry rather than of where you are standing. So does the distance at which it reads 1 W/m^2, at 2.821 m in both: that is a fact about the source, and walking towards it or away from it cannot change it.

The last row is the cheapest check in the whole lab. At one metre the power needed for 1 W/m^2 reads 12.566 W and the sphere area three rows above it reads 12.566 m^2; at two metres they are 50.265 W and 50.265 m^2. They agree because the power for one watt per square metre is the area in square metres, which is the law stated as a fact about geometry. The unit of that area is where the SI units guide starts, and the watt itself belongs to the power guide.

Two: the five sources the buttons load

These are the presets with the readings they produce. Read the last three columns across: two of the five change the cost of doubling the distance, and exactly one of the five has anything wrong with its origin.

Rows two and three differ by one slider and tell the whole exponent story. Same 100 W, same 2 m, and only the shape changed: the point source reads 1.989 W/m^2 and the line source 3.979 W/m^2, which is exactly twice. The cost card goes with it, from 1/4 exactly to 1/2 exactly, and the halving distance from 2.828 m to 4.000 m.

Row four is the one worth loading on its own. A kilowatt at ten metres on the fourth-power rung reads 7.958 mW/m^2 at 0.0001x of the one-metre reading, and the cost card says 1/16 exactly. Sixteen rather than eight, because the echo crosses the sphere twice: that doubles the exponent, which squares the ratio.

Row five is the honest one, and it is the anchor in disguise. Its power, its distance and its shape are row one’s, and its intensity, ratio and cost cards print row one’s strings character for character. Only the last column moved, to 4.0000x. The geometry did not change; the question did.

Formula and Symbol Reference

The band at the head of this page carries the law with the ratio form beside it, because the first on its own needs a source power and the second does not. Every reading on the panel is one of those two forms, a distance that follows from them, or a ratio between two states.

The symbol column below uses the panel’s own names, spelled the way the lab’s own strip spells them in plain ASCII, with pi and rho written out rather than drawn. For which letter belongs to which quantity, the physics symbols guide keeps the list, and the physics formula sheet is the one-page reference this relation sits on.

If you read one row twice, make it the third. Every intensity the panel prints is quoted at a distance counted from the centre of the source, which is not what a tape measure does when the source has a size. The fourth card exists to price that difference, and the row below it is the only place a radius enters the lab at all.

Reading the Diagram: a Growing Sphere, Four Slopes and Two Origins

The drawing is laid out as three bands: the top band carries the sphere, the middle band carries the four laws, and the bottom band carries the two candidate origins. On the largest source-size rung the bottom band draws the source as a disc wide enough to reach up across the middle band, and it is painted after that band rather than under it. The five lowest decade labels, the 0.1 m and 1 m distance ticks and the left part of that band’s own caption lose most of their contrast to it; the bottom band’s caption stays on top, and every reading is in the cards rather than off that axis.

The line count in the top band is held fixed on purpose, and the spacing is what carries the law. The same number of radial lines leaves the source at the detector’s distance and again at twice it, so nothing is lost between the two: what changes is how far apart they are where they cross. Count them and the picture says conservation; measure their spacing and it says fall-off. Drawing more lines further out would have said neither.

All four lines in the middle band pass through one point, and that point is a construction rather than a coincidence. The anchored reading at one metre is byte-identical for all four exponents, which is why the band can carry them on one shared axis and why the label beside that point reads all four agree at 1 m. Only the point-source line is the law; the other three are the same reading given a different slope.

At ten metres the fastest and slowest of those four are 10000x apart, which is what the band’s second caption line states. That is a property of the four exponents rather than a claim about any one engine’s arithmetic: raising ten to the power of minus four lands one bit out in one of the two browsers this lab was compared in, no readout anywhere moves because of it, and the ratio card prints 0.0001x in both. Safari was never exercised at all.

The bottom band’s label is the whole of the fourth slider in one expression. Call the source’s radius R and the gap from its surface to the detector s; then the distance the law actually wants is s + R, and using s instead inflates the prediction by the square of their ratio, which is (1 + R/s)^2. One radius out from the surface that is a factor of four, and the band draws the radius against the gap so the size of the mistake is visible rather than merely stated.

The decade ladders stop rather than skip. The vertical axis labels every third decade, seven labels, at most of the canvas widths tested; at a 380 pixel canvas the stride widens to seven and four labels remain; at a 320 pixel one only the bottom label survives. The distance axis keeps all four of its decades everywhere. Every labelled run is a prefix of the stride grid starting at the first tick, so a reader never sees a gap in the middle of a ladder.

The Physics: Why Squared, and When It Is Not

Leave the panel on its opening state and ask what the sphere in the top band is for. Whatever the source radiates has to cross that sphere, all of it, and the area of a sphere grows as the square of its radius. Divide a fixed power by an area that quadruples and the reading quarters, which is the first two grid rows and the first card in one sentence.

That is also why the exponent is sometimes not two, and the shape slider is the proof. The exponent is a property of the surface the energy crosses and not of the radiation: change the surface and the exponent changes with it, while nothing about the source has moved. The ladder below is that sentence driven through the panel, one row per rung.

Read the second column and the fourth together and the table becomes one idea. A sphere’s area grows as the square of the distance and the cost of doubling is a quarter; a cylinder’s grows as the distance itself and the cost is a half; a flat sheet’s does not grow and there is no cost at all. The echo pays the sphere twice and the cost is a sixteenth, which is the quarter squared rather than halved.

Those four strings are the only four the cost card has. Across all 2,048 states the four sliders can reach it prints 1/4 exactly, 1/2 exactly, 1x -- no change or 1/16 exactly and nothing else, because the cost is exactly two to the power of minus the exponent. A card that printed a decimal there would have hidden that.

The three non-point rungs need their honest caveat, and the panel gives it rather than hiding it. A line source’s intensity really wants a power per unit length and a flat sheet’s a power per unit area, so four absolute scales on one slider would have been four different quantities pretending to be one. Instead all four are pinned to the same reading at one metre, and the first card’s sentence says anchored to the same reading at 1 m whenever the shape is not a point.

Three columns move and one does not. The exponent, the surface and the sentence all change down the four rows; the intensity is the same string in every one, because one metre is where they were pinned together. Note also that only the first row prints an area: the other three print how their surface grows, so no row ever claims a sphere’s area for a cylinder.

Two famous laws wear this exponent for exactly this reason and neither has a slider on this panel. The gravitational force law wants a pair of masses and a constant, and the electrostatic force law wants a pair of charges and a different one. Each is an instance with its own page and its own tools, and naming them is as far as this lab goes.

One clause on sound, and then it stops. The cleared source records that a sound source’s pressure falls off as 1/r while the intensity, which is that pressure multiplied by the particle velocity, keeps the square — one source, one distance, and the exponent depending on which quantity was asked for. Turning either into a level is the decibel formula guide’s job, and there is no logarithm anywhere on this panel.

Where It Breaks Down

Every state the four sliders can reach is answered, and the arithmetic is not where this lab can let you down. What it cannot do is model a real measurement, and what it refuses to do is treat its own idealisations as facts. The first item below is the one that matters more than all the others put together.

Measuring from the wrong origin costs a factor of four

Read this one before any of the others. Every intensity this panel prints assumes the distance was counted from the middle of the source, which is where a spherically symmetric emitter behaves as though all of it sat. Take the gap from its surface instead and square that, and the prediction is too high by exactly (1 + R/s)^2 — 4.0000x one radius out, 1.4400x at five radii, 1.2100x at ten, 1.0201x at a hundred. The one per cent line sits at 200.4988 source radii, which was solved rather than guessed, and 1,792 of the 2,048 states this lab can reach are past it. Adding the radius back before you measure is the whole fix, and the fourth card is there to price not doing it.

That is a different error from an extended source, and the two never correct each other

The cleared source states its own approximation: when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1 per cent. That rule is about treating a source with a size as a point at the right origin; the fourth slider on this panel is about using the wrong origin. They are different quantities, the reference script asserts that they are, and neither may be applied to the other. The source’s rule is quoted here and not extended, because the integral that would extend it needs a model of the source that nothing here provides.

Three of the four shapes are a normalisation, not a second law

Only the point-source rung is P / (4 pi r^2). On the other three the slider changes the exponent while holding the one-metre reading fixed, so the numbers it prints are a fall-off law compared on a shared axis rather than an absolute intensity for that geometry. The first card says so, in those words, on every one of them. If you need an absolute figure for a long tube or a flat panel, this lab is not the place to get it and it does not pretend to be.

The top band is not a drawing you may measure

The gap it draws between the source and the detector is a fixed fraction of the canvas width rather than a length to scale, so the metres per pixel change every time the distance slider moves. Two states cannot be compared by eye in that band at all. The labelled bar is the only thing in it that states a scale, which is exactly why it is mandatory: the distance ladder covers a factor of a thousand. The bottom band is to scale, but at its own scale, so no horizontal distance in one band means anything in the other.

Aim this panel at a beam and the distance slider stops meaning anything

A torch, a spotlight, a dish antenna and a laser all share one problem the cleared source names outright: their effective origin lies far behind the aperture the light appears to leave from. Set a distance on the slider for one of those and you have measured from somewhere that is not where the energy behaves as though it came from, by an amount this model cannot tell you. There is no control for it here and no figure anywhere on the page, because nothing in this cluster sources a beam divergence.

Nothing is lost on the way, which no real medium manages

Absorption and scattering are excluded by assumption here rather than measured away, and the cleared source states its proportionality on exactly that understanding. Air, water, glass and anything else in between take energy out as the quantity travels, so a meter on a bench drops behind this panel faster than the exponent alone would have it. Not one coefficient, half-value layer or dose figure is printed anywhere on this page for any of that, and the gamma ray guide is where attenuation and shielding belong.

Nothing sends energy back across the sphere

The law counts the energy crossing one sphere once. Indoors a reflected path crosses it twice, so a reading taken near a wall or in a small room is higher than this panel predicts and stops falling off at all once the reflected field dominates. No number is attached to any of that here, for the plain reason that no reflection coefficient is sourced anywhere in this cluster.

The detector faces the source and is small compared with the distance

The source defines the reading as what passes through unit area directly facing the source, so a tilted detector collects less and one comparable in size with the distance averages over a range of distances instead of sampling one. Neither is corrected for here and neither has a control. What the quantity itself is, and how it behaves when a detector is not facing squarely, belongs to the polarisation of light guide rather than to this panel.

A rounded distance gets worse when it is squared

The error in a reading is always more than double the error in the distance, by exactly the square of it: one per cent in the distance is 2.0100 per cent in the reading, five per cent is 10.2500, ten per cent is 21.0000, fifty per cent is 125.0000 and a doubling is 300.0000. Nothing on this panel is rounded, because every rung is an exact value, but the moment a distance comes off a tape measure it is. The source’s own quick photographic rules and its solar example are both instances of that, and the written guide works them through.

The wrong geometry gives the wrong exponent, and it is one slider away

Assume the square for a fluorescent tube at close range and the answer is out by the whole difference between 1/r and 1/r^2: at ten times the distance, a tenth against a hundredth. The shape slider is here precisely so that mistake can be seen rather than argued about. The inverse square law calculator has no geometry control at all, by design, so a long, flat or echoing source is this panel’s question and not that one’s.

One sentence in this lab ships and never fires

The sub-line under the fourth card is built to end either outside the one percent rule or inside the one percent rule, and across all 2,048 states the second ending never appears. Seven of the eight size rungs are outside the line — even the rung at one two-hundredth, by 0.0025 of a percentage point — and the eighth is a true point source, which takes the no-error sentence instead. The branch was kept rather than deleted, its reachable count is asserted to be zero, and no reading you can reach here will report being inside that line.

The sliders are ladders, and the lab runs one way only

All four controls step by whole rungs, so the lab reaches 2,048 states and nothing between them. It also runs forwards only: you set a power, a distance, a shape and a size, and read what follows. There is no box to type a reading into and no way to ask for a distance back. Every figure on this page is arithmetic from four slider positions, nothing on it was measured from any real source, and the assumptions it rests on are printed in a fixed line under the drawing.

Where the Inverse Square Law Is Actually Used

Predicting a bench reading before you move the meter

Load the first two presets and the whole decision is on screen in two cards. 7.958 W/m^2 at one metre becomes 1.989 W/m^2 at two, and the grid tells you where the halfway points are: 1.414 m for half the reading and 70.711 cm for double it. Two slider positions and a ratio did all of it, and at no point was the panel told what kind of thing was radiating.

Adding a lamp’s own radius back before you trust a close-range reading

The fourth slider is this case and nothing else. A source whose radius equals the gap to its own surface makes a prediction 4.0000x too high; at a fifth of that gap it is 1.4400x, at a tenth 1.2100x, at a hundredth 1.0201x. The line below one per cent is 200.4988 source radii, so a reading taken close in is almost always over it. The bottom band draws the radius against the gap so the size of the mistake is obvious.

Backing a source power out of one reading

The sixth grid row answers it directly, and it prints 4 pi r^2 at the distance on screen. That is the power for one watt on each square metre on the point-source law: 12.566 W at one metre, 50.265 W at two, and at one metre it is right on all four shapes, because that is where the laws are pinned. It is not that power on the other three rungs away from one metre: the 1.257 kW on the radar preset would put a hundredth of a watt on each square metre at its ten, not one. Scale by your own reading for the total radiated power, provided the thing radiates equally in all directions, which the panel prints as the assumption it is.

Judging whether a source is small enough to treat as a point

This is the question the cleared source itself is sharpest about, and its warning for radiography is that the proportionality does not hold in practical situations unless the source dimensions are much smaller than the distance. The panel offers no correction for an extended source and prints no figure for one, because none is sourced here. What it does give you is the fourth card, which prices the related and commoner mistake of measuring from the wrong place.

Understanding why radar range costs so much more than radar power

Load A radar echo: out and back, 1/r to the fourth. A kilowatt at ten metres returns 7.958 mW/m^2, which is 0.0001x of its own one-metre reading, and the cost card reads 1/16 exactly. Doubling the range costs sixteen times, so recovering it by raising the power means sixteen times the power. The echo pays the sphere twice and that is the entire reason.

Setting a lamp for a one-stop change on a stage or a set

The halving and doubling rows are the working set for it. At one metre they read 1.414 m and 70.711 cm; at two metres, 2.828 m and 1.414 m. The factor is the square root of two in both directions and it scales with wherever you are standing, which is why moving a light a fixed number of centimetres does something different at every distance.

Catching a fall-off claim that is not an inverse square at all

Set the shape slider through its four rungs and read the cost card: 1/4 exactly, 1/2 exactly, 1x -- no change, 1/16 exactly. If a claimed fall-off matches none of those, either the surface is not one of these four or something is absorbing. The cost card is the fastest test on the panel because it needs no units and no second reading.

Telling the law apart from the instances it is famous for

The two best-known cases have their own panels with their own controls: the universal gravitation simulator puts two masses on sliders and the Coulomb law simulator puts two charges on them. Neither control exists here, because this lab is about the surface the energy crosses rather than about what is doing the crossing. Nothing this panel prints can be turned into a force without quantities it was never given.

Where to Go Next

A panel can put you in a state and let you move one thing at a time, and that is the end of what a panel is for. The inverse square law guide is where the subject itself is written out: why the exponent is two, what the ratio form buys you, what four different spreading surfaces do to it, and why measuring from the wrong place costs a factor of four, with the problems worked step by step.

Two posts own the instances this lab deliberately stops short of. The law of gravitation guide carries the gravitational force law with its masses and its constant, and the Coulomb law guide carries the electrostatic one with its charges and its own.

Five more pick up what this lab uses but does not explain. The power guide is the quantity on the first slider, the SI units guide is where the watt per square metre comes from, the decibel formula guide is where a sound intensity becomes a level, the physics symbols guide settles which letter is which, and the physics formula sheet is the one-page reference.

Six more sit one step further out. The electromagnetic spectrum guide is what might be doing the radiating, the sound waves guide and the heat transfer guide are two other things that spread outward, the Stefan-Boltzmann guide is where a radiated power comes from, the Kepler laws guide is where an inverse-square pull becomes an orbit, and the electric field guide is the field version of the same picture.

On the tool side, the inverse square law calculator takes typed readings and distances and solves in four directions, the gravitational force calculator and the Coulomb law calculator are the two instances, the decibel calculator turns an intensity into a level, the Stefan-Boltzmann calculator gives the power a hot surface emits, and the electric field calculator is the field quantity.

Five neighbouring pictures sit beside this one: the universal gravitation simulator, the Coulomb law simulator, the decibel simulator, the Stefan-Boltzmann simulator and the electromagnetic spectrum simulator. Beyond them there is the simulation library, an index of calculators, and the blog for everything written out at length.