Right-Hand Rule Simulator

What Is the Right-Hand Rule Simulator?

How to use the right hand rule physics simulator

1. Read the band above the drawing before you touch a control. It fixes the convention the whole lab depends on: +x is right across the screen, +y is up it, +z is out of the page toward you. A circle with a dot on the canvas is out of the page and a circle with a cross is into it, and all three rules use conventional current. Assume the other sense for z and every answer reverses.
2. Now read the three cards as the lab boots. Force direction reads down (-y), Force magnitude 1.923e-13 N and Angle from v to B 90 deg, with Charge q at +1 e beside them. The strip under the drawing spells the same thing out as a cross product: v x B: right x out of page = down.
3. Press negative (-) and change nothing else. One card moves. Force direction turns to up (+y), Charge q reads -1 e, and Force magnitude holds at 1.923e-13 N — the sign of the charge reverses the force without resizing it.
4. Walk the Field B pad round and watch the force follow. With the velocity left at right, up gives out of the page (+z) and into page gives up (+y), both still at 1.923e-13 N. Those two are worth doing one after the other, because they are the pair most often got backwards on a diagram.
5. Press left on the same pad for the case that has no answer. Force direction stops naming an axis and reads no direction (the force is zero), Force magnitude reads 0 N, the angle reads 180 deg, and the strip says v x B: right x left = zero; v is antiparallel to B. The check card beside it reads 2.355e-29 N, and the fixed line at the foot of the panel says which of the two is the force.
6. Drag Speed v and Field B and watch what cannot move. The speed runs from 0.5 to 10.0 million metres per second in steps of 0.1 million, printed as 3.0e6 m/s at the boot value; the field runs 0.05 to 1.00 T in steps of 0.05, printed as 0.40 T. Both move the magnitude and neither touches the direction card, because the direction came from the pads.
7. Press Straight wire. The moving-charge controls go and the grip rule arrives: Current I runs with the same six words, and From the wire, you go with four. At the boot values — current right, you up, 10.0 A and 0.050 m — the cards read out of the page (+z), 40.00 µT and 0.050 m, and the strip reads I x r: right x up = out of page.
8. Press down on the stand pad and nothing but the direction changes. The field card turns to into the page (-z) while 40.00 µT holds, which is the whole of the grip rule in one press: cross the wire and you cross the field with it.
9. Drag Distance r from 0.050 m to 0.100 m. The field card halves to 20.00 µT and the B at twice that distance cell beside it halves again to 10.00 µT. That second cell is there so the inverse-distance law can be read off without moving the slider at all, which is the one piece of arithmetic on this rule you never have to leave the panel for.
10. Press out of page for the current and look at the stand pad. Its four buttons relabel themselves to right · left · up · down, because those are now the four directions perpendicular to the wire. The lab keeps your slot rather than your word: the first slot was up with the current along x and is right with the current along z.
11. Press Coil and read which end is north. With anticlockwise (from the right end), 2.0 A and 1000 /m, North end reads right end (+x), Field inside 2.5133 mT, and the South end cell left end (-x). Press clockwise (from the right end) and the two ends swap while the field inside does not change at all.
12. Drag Turns per metre n to 2000 /m. Field inside doubles to 5.0265 mT and the north end stays where it was, because turns density sets how strong the field is and the sense of the current sets which way it points. That is the one relation of the three with a calculator of its own: the solenoid magnetic field calculator solves B = µ0 n I for any one of its three quantities, so you can also ask it what turns per metre or what current a field you need would take.
13. Read the caption on the drawing last, and read it as a claim. At the boot setting it says F = q v x B: v right, B out of page, positive charge: F is down, then F magnitude 1.923e-13 N at 90 deg between v and B, then the warning that arrow lengths are indicative, not to scale. The red arrow really is drawn downward, and the caption is checked against the drawing instructions rather than against a flag.
14. Press Reset to the default case when you have lost track. It restores the rule, all three direction pads, the sign, the stand slot, the coil sense and all seven sliders at once. It is the only control that reaches into more than one rule.

The step worth repeating is the third one. Load Proton, field out of the page, then press negative (-) with your eye on the magnitude card. One number out of five changes its sign and nothing changes its size. That is the single most-missed point in the topic, and it takes one button to settle.

For what the rule is — the three forms as physics, the full 36-pair direction table, the four mistakes that account for most wrong answers, and eight worked problems — read the full guide to the right-hand rule. This page is about the panel and the drawing: which card answers which question, and where the picture gives a line up before the numbers do.

Worked example: turn one thing around at a time

Every row below is one setting of the pads and the sliders, and every cell is a string the running lab printed there. The seven moving-charge settings come first; the last column is the check card, which exists so that one row of the table can disagree with itself in public. Where a cell and the lab ever part company, believe the lab.

Rows 1 and 2 are the whole argument of the page in two lines. The same velocity in the same field gives down (-y) for a positive charge and up (+y) for a negative one, at an identical 1.923e-13 N. Nothing about the hand changed between those rows; only the sign in front of the cross product did.

Rows 3 and 4 move the field instead of the charge. Turning the field up throws the force out of the plane of the page altogether, to out of the page (+z); turning it from out of page to into page flips the force from down (-y) to up (+y). Those two rows are why the axis convention is restated on the drawing itself.

Rows 5 and 6 are the pair to look at twice. Both read no direction (the force is zero) and both read 0 N, but their check cards read 0 N and 2.355e-29 N. The second of those is the double-precision sine of 180 degrees, 1.2246e-16, surviving into a product; the cross product returns an exact zero in the same instant. The force is the 0 N, and the strip names which cause applied rather than printing an axis for a vector that has none.

Row 7 is a direction, not a size. Turn the velocity up with the field right and the force goes into the page (-z), at 1.282e-13 N because the speed slider is at 2.0e6 m/s rather than its boot value. Read it against row 1: the same hand, two of the six directions swapped, and an answer that has left the page.

The grip rule, and the only card pair that halves

Switch to Straight wire and the question changes from where a charge is pushed to where the field goes. The first four rows below hold the current at 10.0 A and 0.050 m and move only the two direction choices; the fifth moves the distance instead.

Read rows 2 and 3 against each other first. Standing above the wire gives out of the page (+z) and standing below it gives into the page (-z), at the same 40.00 µT, which is what circulation means: the field has no single direction, only a sense of going round. Row 4 reverses the current instead of your position and gets the same reversal, so two reversals together would put you back where you started.

Rows 2 and 5 are the inverse-distance law on one card pair: 40.00 µT at 0.050 m and 20.00 µT at 0.100 m, with the B at twice that distance cell printing the second figure while the slider is still on the first. This trainer stops at the field and never puts a charge into it; what a field of that size then does to one — Magnetic force from F = qvB, the Radius r of the circle it bends the charge onto and the Period T of that orbit — is what the magnetic field and Lorentz force lab runs as an animation.

The coil, and the one card the sense does not move

The third rule has only two directions to choose between, both read looking at the right-hand end of the coil, so its table is short. What it shows is which card the sense of the current controls and which card it does not.

Rows 1 and 2 swap the two ends and leave 2.5133 mT exactly where it was. Rows 1 and 3 double the turns density and double the field to 5.0265 mT while the north end stays at the right end (+x). So the sense of the current owns the two end cards and the two sliders own the field card, and no setting of this rule lets either group reach into the other.

Formula and symbol reference

The panel prints each of its three relations inside the label of the card it feeds, so no answer arrives without the rule that produced it. F = q v x B sits above Force direction, B = µ0 I / 2πr above Field magnitude, and B = µ0 n I above Field inside. None of the three is derived here.

The hand is assigned differently in each of the three, which is the most common reason a reader who has the first rule right still gets the third one backwards. The table below names what the hand does in each case and which card holds the answer.

The row worth reading twice is the angle. Every other readout on the panel is continuous in the control that drives it; that one can only ever print three values, because six axis directions cannot be at any other angle to each other. A velocity and a field pointing anywhere else are the job of the right-hand rule calculator, which takes six components and gives a unit vector back.

The physics: what the picture can honestly claim

The caption is the part of this drawing to trust, because it is the part that is checked. Its first line states the answer in words, and the arrow and arc instructions the canvas actually received are compared against that line rather than against the variable the drawing branch had just set. If the caption says the force is down, the red arrow was drawn downward.

That is also why the drawing says in its own third line that arrow lengths are indicative rather than to scale. The three arrows of the moving-charge rule are given lengths that fit the box; the force is a hundred-thousandth of the velocity in SI units, so a scale picture of all three would show two of them and a dot. Compare sizes from the cards, and directions from the picture.

The straight-wire rule is drawn two quite different ways, and the difference is a matter of honesty rather than of style. With the current running out of or into the page the field loops lie flat in the page, so the lab draws three concentric circles with six arrowheads on them, going anticlockwise on screen for a current out of the page.

With the wire lying in the page those same loops stand perpendicular to it, and an arrowhead drawn on one would claim an in-plane direction the field simply has not got. So the lab draws rows of circled dots on one side of the wire and circled crosses on the other, each computed from the cross product at that point, leaves the faint perspective loops arrowhead-free, and adds a small inset labelled looking along the current — the one view in which the circulation really is a rotation in the plane of the screen.

What a magnetic field is, and why a charge moving across one travels on a circle at constant speed, belongs to the guide to the magnetic field rather than to this panel. The perpendicularity behind that is on the screen, though: the force card and the angle card together never once describe a force with a component along the velocity.

All three rules read the conventional current, which runs opposite to the drift of the electrons, and the band above the drawing says so before the grip rule or the coil rule is used. Getting that one substitution the wrong way round reverses every field direction on the canvas. The guide to electric current sets out why the convention is the way it is.

Changing a direction, a sign, a sense or the rule regrows the arrows over about a quarter of a second, which is the drawing telling you that the answer moved rather than merely its size. Dragging a size slider starts no animation: the numbers change and the geometry does not. The lab also boots at rest with its arrows at full length, so the first thing you see is a finished picture.

Narrow the window and the canvas gives things up in a fixed order rather than shrinking everything together. Each caption line has a ladder of shorter forms, and a line that will not fit in any of them is dropped from the bottom upwards, so the line naming the answer is the last to survive. The two insets need a wide canvas and are simply absent on a narrow one, which is honest: a picture too small to be read is worse than no picture.

Where the right-hand rule simulator breaks down

The lab solves its own model exactly, so nothing on the screen ever fails. Everything below is a limit of that model, of the six directions the pads offer, or of what a canvas a few hundred pixels wide can carry, and each item says what the lab does about it.

Six directions, and nothing between them

The pads offer right, left, up, down, out of page and into page, and that is the entire input space: 36 ordered pairs for the moving-charge rule, of which 12 give no force at all. The restriction is deliberate, because it is what lets an answer be a phrase rather than three numbers. It also means the Angle from v to B card can only ever print 0 deg, 90 deg or 180 deg, and no setting of the pads reaches anything else.

The arrows are not to scale, and the drawing says so

Arrow lengths on the canvas are indicative. The third caption line states it on the drawing itself, and the two field rules are labelled schematic, not to scale and side view, schematic for the same reason. Nothing on the canvas is a measurement; the cards are.

The straight-wire relation assumes an infinitely long wire

The field card uses B = µ0 I / 2πr, which is exact for an infinitely long straight wire and departs from a real one near its ends. The drawing is honest about this much: the wire runs off both edges of the canvas and is never given a length. Within a few diameters of the middle of a long wire the figure is good; close to a cut end it is not.

The coil relation is the ideal interior of a long solenoid

B = µ0 n I is the value inside a long coil, far from its ends, and the field lines drawn through the bore are the only ones the lab draws for that reason. A short coil has a weaker interior field than this card reports, and the field just outside either end is roughly half the interior value rather than nothing. Where that matters, the guide to the solenoid handles the geometry properly.

The magnetic constant is a measured quantity now, not a defined one

The fixed line prints µ0 = 4π × 10^-7 T·m/A and adds that it has been measured rather than defined since the 2019 revision of the SI, right to two parts in 10 billion. That tightness is the lab’s own figure and is not sourced on this page: how close the agreement comes out depends on which CODATA adjustment you read it against, and a part in a thousand million is the bound that survives any of them. The value itself is used because the other magnetic tools on this site use it, so their figures and these cannot disagree, and it is not called exact anywhere.

Two cards are allowed to contradict each other, once

At an angle of 180 deg the force card reads 0 N and the q v B sin(angle) check card reads 2.355e-29 N. That is not a rounding choice: the sine of 180 degrees in double precision is 1.2246e-16, so the textbook magnitude formula cannot return an exact zero where the cross product does. The force is always the cross product, and the check card is published so the discrepancy is visible rather than hidden.

Nothing moves, so there is no orbit here

The charge is drawn at one instant with one velocity. There is no trajectory, no radius, no period and no animation of the circle the force would bend it onto, because this is a direction trainer and that would be a different lab. The force card is the force now, at the origin, and the next instant is not modelled.

A size from an angle is a different question and a different tool

If you have a speed, a field and the angle between them and you want newtons, you do not need a direction trainer at all. The sine form F = q v B sin(angle) answers it, read with the size of the charge rather than its signed value, and solving that one is the job of the magnetic force calculator — forwards, or back for whichever of the four quantities you are short of. This lab prints the sine form only as a check on its own answer, never as the answer.

Every slider lands on a grid, and every figure is rounded once

The seven sliders are coarse on purpose: 1 e, 0.1 million metres per second, 0.05 T, 0.5 A, 0.005 m, 0.1 A and 100 turns per metre. Nothing between two notches is reachable. Each card is also computed from the exact value and rounded once, so re-deriving one printed figure from another will not always close: read each from the card that publishes it.

The insets and the labels go before the numbers do

Below roughly three hundred pixels of drawing width the two insets are not drawn at all, and the caption falls back through its shorter forms until a narrow column shows only F is down, F magnitude 1.923e-13 N and not to scale. The cards never degrade, so a phone shows the full answer in the panel even where the picture has given most of itself up.

Nothing here has been measured

Two direction choices, a sign and a few slider positions go in, and one idealised answer comes out. A preset name is only a label for the values it writes. No wire, coil, magnet or instrument is described anywhere on this page, and nothing you intend to depend on should leave here unverified.

Where the right-hand rule is actually used

Settling which way a beam will bend, in front of a class

This is the fastest thing the lab does. Set the velocity and the field to match the diagram on the board, read the direction card, then press negative (-) for an electron beam rather than a proton one. down (-y) becomes up (+y) and the magnitude does not budge, which settles the argument about whether the spot lands above or below the axis without anyone drawing a hand on the board.

Decoding the dots and crosses on someone else’s diagram

A field drawn as crosses is going into the page and a field drawn as dots is coming out, and the two are easy to read the wrong way round under time pressure. The canvas prints its own legend, +z out of page, in the corner, and the pad buttons say out of page and into page in words. Set what the diagram shows and let the card tell you what you have actually drawn.

Getting the push on a motor wire the right way round

Replace the charge and its velocity by the current and the length of the wire, and the moving-charge rule becomes the motor rule with the same hand and the same geometry: a wire carrying current to the right in a field into page is pushed up (+y), exactly as row 4 of the first table reports. Set the current direction as the velocity and read the force card.

Deciding which end of a coil will be north before you wind it

The coil rule answers this in one press, and the South end cell beside it removes the second guess. That is the question behind every relay, latch and lifting magnet: which way round to wind it so the pole you need faces the work. The electromagnet lab then shows what a core does to the field the winding makes on its own.

Reading the inverse-distance law off two cells at a glance

The Field magnitude card and the B at twice that distance cell sit side by side, so 40.00 µT and 20.00 µT appear together at one slider position. For a quick judgement about how far from a cable a reading is already negligible, two cells beat solving the relation twice.

Proving to yourself how many pairs give no force at all

Walk the two pads through all six directions each and count: 12 of the 36 ordered pairs give no direction (the force is zero), which is three axes times two senses of the velocity times two senses of the field. Counting six instead is the usual slip, and the pads make the real answer countable rather than arguable.

Teaching that a zero vector has no direction

Most tools print 0 and leave it there. This one refuses to name an axis and says no direction (the force is zero) with the cause on the strip beneath, so the distinction between a force of zero size and a force with no direction stops being a quibble. It is also the honest reason a charge released along a field line simply carries straight on.

Working out which way an induced current will run

Induction is the same geometry read in reverse: the force on the carriers inside a moving conductor is the moving-charge rule, and which way it pushes them is which way the current goes. Set the velocity of the wire and the field it is crossing, and the direction card gives the sense of the current directly. The guide to electromagnetic induction handles the size of what that drives.

Where to go next

The topic itself — what the right-hand rule is, all three forms worked through, the 36-pair direction table, the four mistakes that cause most wrong answers and eight problems end to end — is in The Right-Hand Rule in Physics: Three Clear Rules, One Hand. If your vectors do not lie along the axes, the right-hand rule calculator takes the six components and returns the force as a vector with its working.

The neighbouring questions have tools of their own. The guide to the magnetic field owns how big a field is rather than which way it points, and the solenoid magnetic field calculator turns a turns density and a current into that number. The magnetic field and Lorentz force lab picks the charge up where this trainer sets it down, running it round a circle and reporting the force, the radius and the period, while the magnetic force calculator gives the size of the force on its own.

The guide to the solenoid alongside the solenoid lab and the guide to electromagnets with the electromagnet lab take the coil rule further than one card can.

Further out, the guide to electromagnetic induction and the electromagnetic induction lab, which oscillates a magnet through a coil and reports the flux, the EMF and the current it drives, take the same geometry the other way round, the guide to the Lenz law fixes the sign of what induction produces, and the guide to scalars and vectors with the vector addition calculator is where to go if what you need is a resultant rather than a perpendicular. The other cross product you meet every week is handled by the guide to torque and the torque calculator, and the rest is in the library of physics simulations.