Magnetic Flux Simulator

What Is the Magnetic Flux Simulator?

How to use the magnetic flux simulator

1. Read the four cards before you touch a control. The lab boots paused and square on, so Magnetic flux reads 1.000 mWb, Area facing the field 50.00 cm2, Angle factor 1.000 and Field lines through the loop 5. Each card carries the thing that produced it in its own label, so the headline is labelled Φ = B A cos θ and the second card A cos θ, size only.
2. Leave Flux density B at 200 mT and Loop area A at 50 cm2, and drag Angle from the normal to 60 deg. The flux halves to 0.500 mWb, the facing area halves to 25.00 cm2, the angle factor is 0.500 and two of the five gold lines go, leaving 3. Nothing about the field moved; the grid cell still reads 0.200 T.
3. Carry the same slider on to 90 deg. This is the step the lab exists for. Magnetic flux reaches 0.000 mWb while Area facing the field reaches 0.00 cm2, the angle factor is 0.000, no lines are drawn at all, and Which way it threads reads none. The field is as strong as it ever was and the loop is the same loop.
4. Keep going to 180 deg. The flux comes back as -1.000 mWb, the facing area returns to the full 50.00 cm2, the angle factor is -1.000 and the five lines are drawn again. Which way it threads now reads into the loop: the same size of flux through the same area, arriving at the other face.
5. Now drag Flux density B instead, and watch what refuses to move. At 0 mT the flux reads 0.000 mWb and no lines are drawn, but Angle factor still reads 1.000, because the cosine is a fact about the geometry and knows nothing about the field. Take the slider to 500 mT and the flux is 2.500 mWb at 13 lines, with the facing area unchanged at 50.00 cm2.
6. Drag Loop area A and the flux follows it in exactly the same proportion. At 100 cm2 the flux is 2.000 mWb and the facing area 100.00 cm2; at 1 cm2 they are 0.020 mWb and 1.00 cm2. Flux is linear in the field and linear in the area, and these two steps are that statement with the numbers attached.
7. Press Rotate when you would rather watch than drag. The label turns to Pause and the angle sweeps 0 up to 180 and back at about 30 deg/s, snapped to the same 5-degree grid the slider uses, so every reading that flashes past is one you can also hold still. The dot on the lower plot runs along the cosine curve, through the zero crossing and down to the bottom.
8. Press Reset whenever you have lost your place. It returns all three sliders to 200 mT, 50 cm2 and 0 deg and stops the sweep, which is why every preset button above presses it first. It is the only control that reaches more than one thing at a time.
9. Read the compact grid last, because it is the part that settles arguments. Most it could be gives the flux this field and this loop would carry square on, so you can see at a glance what the tilt has cost; Flux in webers repeats the headline in SI units as 1.000e-3 Wb; and Angle prints 0 deg from the normal with the line it is measured from named in the value itself.
10. Read the caption under the drawing as a claim about the drawing. Square on it says loop square to the field - maximum flux; edge-on, edge-on - no flux threads the loop; past 90 degrees, flipped past 90 deg - the flux reverses. Under it sits each line drawn through the loop is 0.2 mWb, and the gold lines really are drawn in the number the card reports.

The step worth repeating is the third one. Load Loop square to the field, then drag the tilt to 90 deg with your eye on the grid rather than on the headline. 0.200 T does not budge while 1.000 mWb becomes 0.000 mWb, and that is the whole difference between a flux and a flux density on one screen. If you want typed figures rather than the 5-degree grid — 35.5 deg, say, or a field well past this slider’s top end — the magnetic flux calculator takes the same three quantities as boxes and prints the substitution.

For what a flux is — the definition, the weber written three ways, the full geometry in prose, why the net flux out of a closed surface is zero and seven worked problems — read the magnetic flux guide. This page is about the panel and the drawing: which card answers which question, which slider can and cannot move each of them, and where the picture gives up before the numbers do.

Worked example: change one thing at a time

Every row below is one setting of the three sliders, and every cell is a string the running lab printed there. The first four hold the field and the loop still and move only the tilt; the fifth changes both of the other two. If one of these cells and the running panel ever disagree, the panel is the one to trust.

Rows 1 to 3 are the argument of the page in three lines. The first two columns never change, so whatever happens to the last column is the tilt’s doing and nothing else’s. The angle factor and the facing area fall together, in step, from 1.000 and 50.00 cm2 through 0.500 and 25.00 cm2 to 0.000 and 0.00 cm2.

Row 4 is the one that surprises people. At 180 deg the facing area is back at 50.00 cm2 — the full area, exactly as in row 1 — while the flux reads -1.000 mWb. That is deliberate: the facing-area card reports a size, so it cannot tell you which face is doing the facing, and the sign is carried by the flux card and by the sense cell instead.

Rows 2 and 5 are the pair to read against each other. Both print 0.500 mWb, and they get there by routes that have nothing in common: row 2 by halving a full loop with a tilt, row 5 by putting a fifth of the area into a field two and a half times as strong, square on. Their angle factors are 0.500 and 1.000, and Most it could be separates them outright at 1.000 mWb against 0.500 mWb. A flux reading alone cannot say which of the two you are looking at.

One pair in this table does not mean what it looks like. Take Flux density B to 0 mT with the loop square on and the flux reads 0.000 mWb while the angle factor reads 1.000 — the same angle-factor cell as row 1, beside the same flux cell as row 3. Neither follows from the other: the factor is the cosine of the tilt and the flux is the product of all three quantities, and a field of zero takes the product to nothing without touching the geometry. The field itself is what the guide to the magnetic field covers.

Formula and symbol reference

The panel prints the relation inside the label of the card it feeds, so an answer never arrives without the rule: Φ = B A cos θ sits above Magnetic flux, A cos θ, size only above the facing area and cos θ above the angle factor. There is no physical constant anywhere in it, which means every figure the lab reports is exact arithmetic on three slider values.

The last column below is each slider’s own end stops rather than the range the physics allows, and that distinction matters when a real field runs off the end of one. The field slider stops at 500 mT, which is half a tesla, so any field stronger than that has to go to the magnetic flux calculator, whose field box takes tesla directly and has no top end. For the field a long winding produces in the first place, the solenoid magnetic field calculator solves B = µ0 n I for any one of its three quantities — it has no area and no angle, so its answer is the number you then bring to this panel.

The compact grid under the four cards is there to hold the quantities a reader has to compare, in the order the build settled on: the four numeric cells first, then the two whose values are sentences and therefore get a full-width row each rather than being cut off halfway through their own claim.

Two of those cells are worth a warning. Most it could be is a product and divides nothing at all, so it is immune to every edge the rest of the panel has to guard against; and Flux in webers prints 0.000e+0 Wb at zero, with one exponent digit and no sign, because the flux card never prints a negative zero.

The physics: why the drawing narrows and the field does not

The upper band of the drawing is not a side view and not a plan view. The loop is drawn as the field sees it — looking straight along B — and the note in the corner of the canvas says so in as many words: field uniform, pointing right; loop drawn as the field sees it.

That one choice is what makes the shading mean anything. In that view the shaded face narrows in exactly the ratio the facing area falls by, so the width you can see is the quantity the second card reports. Drawn in perspective the shading would be decoration, and the narrowing to a bare vertical line at 90 degrees would look like an accident of the angle rather than the point of the lab.

The field arrows are the other half of the argument, and they never change. They stay evenly spaced, the same length and pointing the same way whatever the three sliders are doing, because a uniform field is the model. Every one of those arrows is still crossing the space the loop sits in when the flux card reads 0.000 mWb; what has gone is not the field but the opening presented to it.

The dashed vertical line through the middle of the loop is the axis it pivots about, and it is drawn so the turning is readable as a rotation rather than as the loop shrinking. The short arrow out of the centre labelled normal is the line the angle is measured from, and the arc labelled theta appears between it and the field at every setting but 0 deg, where there is no angle to mark.

The lower band plots the flux against the tilt over the whole 0 to 180 range, with a zero line and a filled dot at the current angle. Its two end ticks are not fixed: they are Most it could be and its negative, so they read 1.000 and -1.000 at the boot setting and 0.500 and -0.500 for the stronger field through the smaller loop. With the field slider at zero there is nothing to scale by and the plot prints no end ticks at all, with the dot sitting on the zero line.

Its horizontal axis is titled theta (deg) - angle from the normal where there is room for it, and simply theta (deg) where there is not, which on a phone is what you get. Spending the slack on naming the line the angle comes from is deliberate: it is the single thing this topic goes wrong on, and the axis of the graph is the last place a reader looks before they believe the shape of the curve.

The curve itself is the best argument on the page that the cosine is not a fudge factor. It is flat near 0 degrees, so the first few notches of tilt cost almost nothing — the angle factor is still 0.996 at 5 deg and 0.966 at 15 deg. It is steepest at the zero crossing, where five degrees either side of edge-on is the difference between 0.087 and -0.087. A sensor a few degrees out of alignment barely notices; one near edge-on notices everything.

Where the magnetic flux 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 grids the three sliders move on, or of what a canvas a few hundred pixels wide can honestly carry.

The field is uniform everywhere and never runs out

Watch the arrows while you drag any of the three sliders: they keep the same length, the same spacing and the same direction right across the band, and nothing the loop does disturbs one of them. That is the model stated on the canvas, and it is a fair one inside a long winding or across a small patch of a large magnet.

It is a poor one for a wide loop held close to a small magnet, where the field at the loop’s rim is weaker than at its centre and leaning a different way as well — and a single Flux density B slider has no way of saying so. The panel will hand you a figure regardless; what it cannot do is warn you that one number was never going to be enough.

The loop is flat and rigid, so it has exactly one normal

There is one angle slider, so there is one normal arrow on the drawing, and that is only honest for a surface whose face points the same way all over. Bend or fold it and the arrow would have to be drawn in a different direction at every point of the rim, with its own share of the flux counted separately — no setting of these three sliders can reach that. The loop never deforms either: dragging Loop area A grows and shrinks the circle on screen without ever creasing it.

The drawn lines are a counting aid, not a count of anything

Each gold line through the face stands for 0.2 mWb, and the card reports the flux divided by that and rounded to a whole number, which runs 0 to 25 across the slider box. Field lines are a drawing convention and not objects, so nothing is being tallied. What the count is good for is noticing a change you would otherwise have to read: three lines instead of five is a halving you can see from across a room.

Nothing changes with time, so no voltage is computed anywhere in this lab

There is no time control, no rate of change and no EMF readout, and the Rotate button is a convenience for sweeping the angle rather than a demonstration of a changing flux: it snaps to the slider grid precisely so that every frame is a setting you could have dialled in and held. A flux that genuinely changes drives an EMF round a circuit, which is Faraday’s law, and the Faraday's law formula guide owns that relation. The lab for it is the electromagnetic induction simulator, whose controls are a drive frequency, a turns count, a field and a coil area — and no angle at all, which is the gap this one fills.

The three sliders land on grids, and nothing between two notches is reachable

The field moves in 10 mT steps, the area in 1 cm2 steps and the angle in 5 deg steps, which is coarse on purpose: every reading the lab shows is one you can get back to, and that includes every frame the sweep flashes past. It also means the readings here are a sample of the cosine rather than the whole of it. A tilt of 35.5 deg, or any field past 500 mT, belongs in a box rather than on a slider.

The facing-area card cannot tell you which face is facing

It reports a size, so 50.00 cm2 at 0 deg and 50.00 cm2 at 180 deg are the same reading for two opposite situations, and 25.00 cm2 appears at both 60 deg and 120 deg. That is the honest choice — a card reading minus fifty square centimetres would raise a question about negative areas that has no good answer — but it does mean that card alone never settles a sense. Read the sign off the flux card and the words out of the sense cell.

The sign is a choice you are being shown, not a direction in space

Out of the loop and into the loop are statements about the normal the lab has drawn, which is the one pointing out of the shaded face. Flip that choice and every sign the panel prints flips with it, without one thing about the field having changed. A worked answer that comes out the other way up has usually started from the other face, which is a disagreement about labelling and not about physics. Which way an induced current then runs is a different question again, and the Lenz's law guide is where it belongs.

The canvas gives words up before it gives numbers up

Narrow the window and the caption lines fall back through shorter forms in a fixed order, so loop square to the field - maximum flux becomes square to the field: maximum flux and then maximum flux, while the axis title drops to theta (deg). A line that will not fit in any of its forms ends the strip rather than being skipped, so what survives is always a contiguous run from the top. The four cards never degrade at all, so a phone shows the full answer in the panel even where the drawing has given most of its labels away.

A coil of several turns is a different quantity, and this lab has no turns control

Winding the wire round the same hole twice does not make more field pass through it, so the flux is unchanged; what changes is the quantity you get by multiplying it by the number of turns, which the Faraday's law formula guide calls the flux linkage. There is deliberately no turns slider here, because a flux belongs to the surface rather than to the wire. Take the figure off this panel and multiply it yourself if that is what your problem wants.

Nothing here has been measured, and two of three engines were tested

Three slider positions go in and one idealised figure comes out; a preset name is only a label for the values it writes. No magnet, core, coil or instrument is described anywhere on this page, and no measured field, geometry or material appears, because no source for one was fetched for this cluster. The readouts were compared in Chrome and Firefox and agreed at every width checked; the WebKit build on that machine could not be launched, so nothing here is claimed about Safari.

Where magnetic flux is actually used

Setting a core’s working point on two sliders

A transformer or motor designer carries two figures at once: the total flux the magnetic circuit has to pass, in webers, and how hard that flux is being pushed through each square metre of steel, in tesla, because past a certain density the iron saturates and stops helping. Those are the headline card and the Flux density B cell, and the Loop area A slider is the cross-section between them. No saturation figure is quoted on this page, because the density steel gives up at is a property of the particular material and its working conditions rather than of this relation — verify before use.

Costing a sensor’s misalignment in one drag

A flux-gate or search-coil sensor responds to the field along its own axis, so mounting it a few degrees off costs you the cosine of the error. Drag the angle one notch from square on and Angle factor reads 0.996; three notches, 0.966. That is the practical form of a result the curve makes obvious: alignment is forgiving near square on and brutal near edge-on, where the same five degrees moves the factor from 0.087 to -0.087 and takes the sense with it.

Sizing the flux through a small head or pickup

A tape head, a stripe reader or a search coil is a small loop with a field through it, so its reading is set by the same three quantities and by nothing else. Take Loop area A down to 1 cm2 and the lab is working in the band those devices live in: at 200 mT square on the flux is 0.020 mWb, which is twenty microwebers. The arithmetic is this panel’s; the figures for any particular device are not ours to quote — verify before use.

Settling the sign question in front of a class

This is the fastest thing the lab does. Load Flipped right over and the flux is -1.000 mWb while the facing area is the full 50.00 cm2, with the sense cell reading into the loop in words rather than leaving a minus sign to be interpreted. Then point out that the only thing making it negative is which face the drawn normal came out of, which is a choice and not a measurement.

Talking about a scanner magnet without pretending to measure one

An MRI magnet is specified by its flux density, and the flux matters for everything that sits in it with a surface: a receiver coil, a screening panel, a steel trolley. The geometry is exactly this panel’s — a field, an area and an angle from the normal — but the field itself is past the top of this slider, so the lab can show you the shape of the answer and not the size of it. No figure for any particular scanner appears on this page, because none was sourced for it — verify before use.

Auditing a flux you already have

The commonest use of a panel like this is checking a figure rather than producing one. Any flux larger in size than Most it could be is wrong before the working is even read, because the angle factor can never exceed 1.000 in either direction. Set the field and the area a problem gives you, read that cell, and compare: it is a one-line sanity check that catches an upside-down cosine and a square-centimetre area left unconverted.

Getting a genuinely uniform field to feed in

The model assumes one, and the usual real source of one is the inside of a long winding, where the field is very nearly constant over the middle. The solenoid simulator is where that field comes from — it reports the field inside the coil and the flux through one of its turns, with no surface to tilt — and the figure it gives is the one to bring to the Flux density B slider here.

Seeing why anyone wants the number at all

A steady flux does nothing by itself, which is a fair question to raise about a panel that computes one. The answer is that a flux which changes drives a current round a circuit, and the electromagnetic induction guide covers that phenomenon end to end. What this lab is for is the ingredient: getting the geometry of the flux right before anything is allowed to change.

Where to go next

The topic itself — what a flux is, what the weber measures, the geometry in prose, what happens past 90 degrees, why the net flux out of a closed surface is zero and seven problems end to end — is in Magnetic Flux: Formula, Units and Worked Examples. When you need figures the sliders cannot reach, the magnetic flux calculator takes any three of the flux, the flux density, the area and the angle and returns the fourth with the working.

The field side of the story has tools of its own. The guide to the magnetic field covers what a flux density is and where it comes from, the solenoid magnetic field calculator turns a turns density and a current into one, and the magnetic field and Lorentz force lab puts a moving charge into a uniform field instead of a surface. The guide to the solenoid with the solenoid lab is where a uniform field actually comes from, and the guide to electromagnets takes the same winding further.

Downstream of this page, the Faraday's law formula guide and the guide to electromagnetic induction pick the story up at a flux that changes, with the electromagnetic induction lab for the moving case; the Lenz's law guide and the Lenz law lab fix which way the result runs, and the guide to eddy currents is what happens when the conductor is a sheet rather than a loop. The rest is in the library of physics simulations.