# What area does the flux formula refer to?

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I've been studying em induction and in my book it was explained by considering a metal rod of length l moving through a magnetic field and cutting through the field lines at a constant speed. So in time dt it moves through ds and they showed e=BA/dt, where A = l x ds(the area that the rod "sweeps out".

Then in the next chapter it explains induction when moving a bar magnet through a coil and this time the "A" in the formula referred to the area of the cross section of the coil (the circle area of one of the rings of the wound up wire). Why is this? The magnetic field lines are only cutting through the small area of the wire, the rest of the "circle" is empty space so why are we considering the whole area of the cross section. What's different now that we're not looking for area that the wire "sweeps out"?

Thank you for your help

Then in the next chapter it explains induction when moving a bar magnet through a coil and this time the "A" in the formula referred to the area of the cross section of the coil (the circle area of one of the rings of the wound up wire). Why is this? The magnetic field lines are only cutting through the small area of the wire, the rest of the "circle" is empty space so why are we considering the whole area of the cross section. What's different now that we're not looking for area that the wire "sweeps out"?

Thank you for your help

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If you have a rod sweeping out an area dA in time dt, this is the same as the rod of length L moving a distance ds in time dt. Given that the rod moves perpendicular to the field, then the area swept out is L ds = dA. It's the same as picking up a piece of lead graphite and scraping it on the edge, you form a rectangle.

As for the bar magnet, emf = -dN(phi)/dt ; the emf induced is proportional to the rate of change of flux linkage through the coils. Flux density, B, is the flux per unit area, so flux is just the quantity of itself. Flux linkage is the total amount of flux through one loop multiplied by the number of loops that flux passes through; N(phi). At A-Level you assume that the flux 'fills' all space within the loops. Obviously, if you think of the extreme such as a 20 mile radius loop with a small bar magnet in the middle, the flux is not going to fill the whole area of that loop.

But for sufficiently small loops, it is safe to assume that the flux from the magnet fills the whole area of the loop/s.

As for the bar magnet, emf = -dN(phi)/dt ; the emf induced is proportional to the rate of change of flux linkage through the coils. Flux density, B, is the flux per unit area, so flux is just the quantity of itself. Flux linkage is the total amount of flux through one loop multiplied by the number of loops that flux passes through; N(phi). At A-Level you assume that the flux 'fills' all space within the loops. Obviously, if you think of the extreme such as a 20 mile radius loop with a small bar magnet in the middle, the flux is not going to fill the whole area of that loop.

But for sufficiently small loops, it is safe to assume that the flux from the magnet fills the whole area of the loop/s.

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What matters is the total flux that penetrates a given area (represented as the greek letter phi), rather what portion of that area is actually penetrated by the flux (which is flux density, denoted by B, unit Tesla).

B*A = phi. So the first formula says that the induced potential difference is equal to the rate of change of the total flux - when the metal rod is moving, it changes the total flux because the area increases. This situation is normally considered where the rod is the 4th side of a closed electrical loop, and that that makes it a bit more clearer visually.

These are two different scenarios that you are asking about. In the first, the area changes. In the second, the flux density changes.

I don't know if that answers your question but I hope it helps.

B*A = phi. So the first formula says that the induced potential difference is equal to the rate of change of the total flux - when the metal rod is moving, it changes the total flux because the area increases. This situation is normally considered where the rod is the 4th side of a closed electrical loop, and that that makes it a bit more clearer visually.

These are two different scenarios that you are asking about. In the first, the area changes. In the second, the flux density changes.

I don't know if that answers your question but I hope it helps.

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(Original post by

I've been studying em induction and in my book it was explained by considering a metal rod of length l moving through a magnetic field and cutting through the field lines at a constant speed. So in time dt it moves through ds and they showed e=BA/dt, where A = l x ds(the area that the rod "sweeps out".

Then in the next chapter it explains induction when moving a bar magnet through a coil and this time the "A" in the formula referred to the area of the cross section of the coil (the circle area of one of the rings of the wound up wire). Why is this? The magnetic field lines are only cutting through the small area of the wire, the rest of the "circle" is empty space so why are we considering the whole area of the cross section. What's different now that we're not looking for area that the wire "sweeps out"?

Thank you for your help

**mrno1324**)I've been studying em induction and in my book it was explained by considering a metal rod of length l moving through a magnetic field and cutting through the field lines at a constant speed. So in time dt it moves through ds and they showed e=BA/dt, where A = l x ds(the area that the rod "sweeps out".

Then in the next chapter it explains induction when moving a bar magnet through a coil and this time the "A" in the formula referred to the area of the cross section of the coil (the circle area of one of the rings of the wound up wire). Why is this? The magnetic field lines are only cutting through the small area of the wire, the rest of the "circle" is empty space so why are we considering the whole area of the cross section. What's different now that we're not looking for area that the wire "sweeps out"?

Thank you for your help

The fundamental principle involved with em induction is contained in Faraday's law. Have you done this?

Faraday explained that you can calculate "rate of change of magnetic flux" in these two ways, depending on which situation you wanted to look at.

The flux passing through the end of a coil = BA if the flux is normal to the coil, A is its area, and B the flux density of the field. This value is for every "turn" of wire, so if there are N turns the so-called flux linkage is NBA.

Any change in this value constitutes a change in the flux linking the coil and will generate an emf in it. The emf will depend on the rate of change of this flux linkage.

So for example, if you were to move the coil completely out of the field in 1 second, the flux linkage would change from NBA to zero in one second. The emf would have a value depending on the values of B, A, and N.

Faraday also said that in order to move the coil out of the field, the edges of the coil have to move through the field and "cut" the lines of flux. This, he argued, was another way of understanding how an emf is generated, by considering the wires themselves cutting through the field rather than the area of the coil.

For example, if you rotate a coil, you change the area facing the field, and so change the effective vale of "A" in the formula. So you get an emf from rotation by changing A. On the other hand, to rotate the coil, its edges cut through the field as they move.

Faraday showed that the emf can be calculated either by considering the flux linking the area, or by considering the flux cut by the edge of the coil or by a single wire. In the case of a single wire, you find the area it sweeps out as it moves through the field.

Both methods are really just mathematical constructions to enable you to calculate the induced emf.

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