# Finding the change in flux

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How do I find the d{flux}/dtime in the faradays law? The problem i'm facing i need to flux which depends on B and area. I have a coil of wire which is about 6 inches away from the center which hold a permanent magnet. How do I find the B [flux density] in the coil wire and how do find the flux change with respect to time? I know the how fast the magnet is moving straight up but i have NO idea how to find the B-flux lines passing through the wire.

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3. ### Blibbler

66
I think you'll find this interesting:

If you're trying to find a numerical solution to a particular problem then the wiki article will show the right equations; if you're asking someone here to {do your homework for you} give help on how to apply the equations then you need to give all the info in the problem set for you.

18
its more of a project, i read hte faradays wiki page, the page doesn't mention anything about measuring change in flux density as you go away from a permanent magnet. I need to find the best line of fit by plotting the change in flux with distant and derive it with respect to time to find d{flux}/dt = induced voltage

18
all it has is B(r,t)... what i have is magnet with all the properties known radial distance to the coil of wire, about 9 cm. I need to find the amount of flux on it. I also need to find the flux on the wire when the manget moves vertically up with 1cm increments.

I have no clue how to do it. Do I need a simulation software?it? Is there similar equation to force of gravity one? Like the further away you are, the lesser force of between two masses which is inversely proportional to square distant?

I know flux density isnt constant everywhere!

6. ### Blibbler

66
Yes, the equation is of exactly the same form as the gravity equation.

18
how can tell me flux density over a specific area from r distance from the source of magnetic flux?

8. ### omoplata

315
I don't know if I understand your question correctly, but it seems like you need the $$\vec{B}(\vec{r},t)$$ for your permanent magnet so you can calculate the flux density that goes through your coil of wire.

I've never seen an equation for the $$\vec{B}(\vec{r})$$ of a permanent magnet in physics problems. All I've seen is equations for the $$\vec{B}(\vec{r})$$ of electromagnets, which can be found using the Biot-Savart law. But I GUESS you can find one for a permanent magnet if you assume it to be a lattice, find the magnetic dipole moment $$\vec{\mu}$$ of the typical lattice point (atom or molecule), find the $$\vec{B}(\vec{r})$$ for a generic $$\vec{r}$$ (this would be the magnetic field due to a single atom/molecule), and spatially integrate over the structure of the lattice to account for all the lattice points.

Maybe these links will help if you want to do that.
http://en.wikipedia.org/wiki/Magnetic_dipole_moment#Magnetic_moment_of_an_atom
http://en.wikipedia.org/wiki/Perman...r_magnets:_magnetic_poles_and_atomic_currents
http://en.wikipedia.org/wiki/Magnetic_field#Magnetic_field_of_permanent_magnets
http://en.wikipedia.org/wiki/Dipole#Field_of_a_static_magnetic_dipole
http://en.wikipedia.org/wiki/Magnetic_pole_strength#Magnetic_Coulomb.27s_law

But since you said it's a project, maybe you have the physical material handy. I'd take an experimental approach in that case, and measure the induced voltage of the coil of wire and the position of the magnet at the same time. Then from Faraday's law of induction, the induced voltage is proportional to $$\frac{d \phi}{d t}$$ where $$\phi$$ is the flux.

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Thanks for link. I never knew the magnets were anything like 'dipole'. After going throught the pages, I think found the equation I was looking for.

Just to clear myself to you what I'm trying to do. I have a coil of wire in a rod shape around a magnet, not a simple coil around a magnet. The circular rod is filled with N number of wires. The diameter of the coil is known and the distance between the coil and center magnet also known. I just want to find the amount of flux entering the surface area of the wire.

Anyway, returning to the equation, I don't get it. :'(

The type of magnet is known, but how do I find its dipole movement? What the difference between 'lamba' and 'r'? Aren't the the same thing distance from the center axis of the magnet?

If 'p' is the distance from the z-axis... wut is 'r' then??????

10. ### omoplata

315

It looks like the $$r$$ comes from the spherical coordinate system, and $$\rho$$ comes from the cylindrical coordinate system. $$\lambda = 90^{\circ} - \theta$$ is an angle which can be calculated using the spherical coordinate $$\theta$$

$$\rho$$ is the distance from the $$z$$ axis, and $$r$$ is the distance from the origin. From the Pythagoras's law, $$r^{2} = \rho^{2} + z^{2}$$.

I don't know if a whole magnet could be accurately approximated by a single dipole moment. But it might be possible. What I meant in the last post was to take in to account all the molecules in the magnet, and assign each of them a magnetic dipole moment, and then calculate the total magnetic field from all of those molecules. But reading about this further, it seems more complicated than I thought. I don't know how to find the individual magnetic dipole moments of molecules (my quantum mechanics is not that good at the moment). The temperature has to be taken in to account too. The Atomic, Solid State, Comp. Physics forum might be a good place to ask how to model a permanent magnet. Solid state physicists do that stuff for a living.

Like I said before, you could also take an experimental approach. If you can measure the voltage between the two ends of the coil at any given time $$t$$, then that is equal to the electromotive force $$\varepsilon$$ in the Faraday's Law, $$\lvert \varepsilon \rvert = N \lvert \frac{ d \phi_{B} }{ d t } \rvert$$, where $$N$$ is the number of coils. So you can measure the rate of change of flux.

You can integrate this(you could do it numerically) to find the flux $$\phi_{B}$$. But there will always be an unknown constant. One suggestion I have to overcome this is start with the magnet very far away from the coil so the initial flux is zero ($$\phi_{B 0} = 0$$) so that constant is zero, and measure $$\varepsilon$$ with $$t$$. If you also record the position of the magnet with time, then you can find out how the position of the magnet is related to the total flux through the coil.

By the way, I found this cool video of a fluxgate magnetometer.

May I ask why you need to find the flux through the coil?

18
Its really late here, going to re-read your post and the equatino. I'm going to assume loads of values but I need to use the temperature too. Going to assume its dipole and everything. I just want to find flux density on a flat small simple xy plane section!,

its just for a simple project. i really don't want to make a practical of it..

12. ### omoplata

315
It looks to me as if finding the $$\vec{B}(\vec{r})$$ for a permanent magnet theoretically would be extremely difficult because it depends on the atomic properties of the magnet. That means quantum mechanics, oscillation of the atoms due to temperature, interactions of atoms with each other, all come in to play. Ising model was an attempt do this, but it is EXTREMELY more simplified than real life. To my knowledge, there is still no analytical solution for the 3D case. You can write whole scientific papers about it and still have no solution.

Finding the $$\vec{B}(\vec{r})$$ for simple electromagnets (coils, solenoids etc.) theoretically are comparatively simple because it only depends on the magnitude and spatial distribution of the current going through the electromagnet.

So I think unless you can replace your permanent magnet with an electromagnet, you have to do a practical experiment to find out the flux through the coil.

18
How about we assume the permanent magnet has the same properties of an electromagnet. The permanent magnet which I chose was Neyodium Iron Boron magnet with its magnetic properties known. If we assume the permanent magnet behaves like a electromagnet, how will I then calculate the flux density on a specific area constant area as the magnet moves up and down?

My main goal is to find the amount of current induced in the coil of wire with a velocity equation name. The whole thing is simple faraday's law induction experiment but instead of the coil surround the magnet, the coil is more wrapped in a circular rod fashion as shown in the figure with it moving vertically straight up at, lets say 10m/s.

14. ### omoplata

315
The $$\vec{B}$$ for an electromagnet depends on the current. For example, look at the equation for a solenoid. It changes with the current $$i$$. There is no analogous parameter for a permanent magnet.

18
I think i found what I was looking for take a look at it. http://www.magneticsolutions.com.au/magnet-formula.html#sr

it doesn't involve any dipole things you mentioned.

>>I THINK THAT WAS IT<< but I'm trying to understand how the equation work. Like I want the field on the coil of wire with a plane perpendicular to the plane of screen [into the screen].

To give you a better model of my setup, it involves two magnet,side by side, surrounded by steel, then the coil of wires. below is the front cross section of the diagram.

AFter going through the page, this is the equation which I see is the best for my setup, I will assume everything is in a block of steel with no air gaps or anything

Couple of questions popped up
• The equation will give me flux density in a rectangular region which is about has a distance of X from the center of the magnet
• I can alter my design to have the coil of wire in a square like region instead of circular cross section has depicted in the diagram
• Assuming I have calculated the flux density on the face of the cross section of the square region which will house the coil of wire, how will I exactly calculate the flux change as the magnet moves up or down?
• More specifically, how will be able to caculate flux change w.r.t to time ot find the voltage induced? If I find an equation for flux density change w.r.t time, I might be able to simple graph it and find the equation of best line of fit using excel. This equation could be derviced w.r.t t and DONE. fLUX change w.r.t time. I hope I am not wrong.

Because the equation doesn't have anything related to vertical distance, I can maybe use simple trig to cacluate the x component of distance as the magnet moves up [x1 and x2]. Though not accurate, it will be able to provide me with some results

Or did I understand the whole equation wrong? :'(

Last edited: May 15, 2011
16. ### omoplata

315
Nice! Thanks for the link. Maybe I can use it sometime.

Use of their equation require a $$B_{r}$$. Do you know this value for your magnet?

I mentioned magnetic dipole moments because I was thinking about calculating the magnetic field using the atomic structure of a magnet. The first three paragraphs on this page from their site gives a brief overview about the structure of a magnet.

It would be great to know how they found out those equations. They haven't posted about it.

18
Last edited: May 15, 2011

18
Btw, is it possible using FEM software? I have ansys but I don't know how to use it yet. I do have CAD model of whole setup. Maybe if you know, I can forward you the model to carry out the FEM. I don't have time right now to learn whole Ansys.

If I'm able to do what this author as done http://img687.imageshack.us/img687/8389/11yatchevilievahinov.pdf it will save me loads of time understand the equations and caclculating the time. But I find it more interested to actually solve all the equation and assume necessary conditions.

19. ### omoplata

315
Tell me if my interpretation of your diagram is incorrect. The two blue rectangles are the magnets. The three white rectangles (with one between the two blue rectangles and two outside them) are steel. I guess the purple squares are air? And the brown circles are your wires.

And if I also understand correctly you intend to change the setup of your magnet to the one they have given the equation for.

The equation they have given is valid for a point along the central axis of the magnets. But I GUESS you can approximate that for the whole rectangular area between the magnets IF you have the magnets VERY CLOSE TOGETHER.

Then if you can move the magnet(or move the coil), so the coil passes between the rectangular space between the two magnets, and keep the plane of the coil perpendicular to the central axis of the magnets, the flux through the coil $$\phi_{B}$$ is $$\phi_{B} = B \cdot A$$, where $$B$$ is the flux density and $$A$$ is the area of the coil that lies in the rectangular space, right?

So $$\frac{ d \phi }{d t} = B \frac{d A }{d t}$$.

If the magnet is rectangular and the length of the magnet is INSIDE THE RECTANGULAR SPACE is $$l$$ (the coil is moving relative to the magnet in the direction of $$l$$), and the width of the magnet is $$w$$, then $$\frac{ d A}{d t } = w \frac{ d l }{ d t }$$.

So $$\frac{ d A}{d t }$$ will be positive as the coil enters the space between the two magnets and negative as it exits the space between the two magnets (The direction of the voltage will change). The exact equation will depend on how the magnet or the coil moves (will it be falling under gravity or moving at a constant velocity or etc...).

They're very clear that the equation is for the distance along the central axis of the magnet. So going off axis will not work. But if you keep the magnets very close together, there will be an almost constant flux density between them, but there will be almost none outside. So the coil will suddenly enter an area of non-zero constant flux density.

I have no expertise with Finite Element Method or Ansys.

18
Thanks I think the problem is solvable,. I want to be sure I'm on the right track

First, the magnet is in a small air pocket, then comes the steel/iron which has the coil in it. I exaggerated the diagram.
Second, it will be only the magnet moving vertically up and down with a defined velocity equation just like in how induction works but instead of coil of wire surround the magnet it will be in a circular rod shaped all tightly winded up together.

The brown is the coil of wire going around the two magnets which will be stationary
The gray region represent the steel or iron, haven't decided which one, probably going to go with iron
the light blue region represent air

So coming back to the equations, lets say the magnet moves up,

Lets say the cross section area of wire from the front view is the yellow region, the flux density region created by the magnet is the red region. Initially the coil of wire lies in the region of the flux but when the magnet moves up, the flux region too moves up producing change in flux region in the coil of wire. I kept the flux perpendicular to the motion of the magnet and assumed if the magnet moves up, the flux region also moves up instead of creating a new flux which will be lower in strength which would still exist, I said, it will not exist..

Coming back to equation, the magnet is rectangle, the magnets are close together too.
1. I will calculate the using the equation from the link by simply plug n chuging the numbers
2. Next I will find the flux area changing at intervals, time:1,2,3,4 ... the veocity motion is periodic and magnet will come down. I will plot it and find the equation of flux change w.r.t. So the area will change with velocity of the magnet moving up, the faster it moves up, the more flux change there will be in one second, the more current will be induced.
3. Now I have B and area change w.r.t, I will be able to find voltage induced
4. i'm going to assume the air pocket to be very samll so the affect will be negilible but practically, the air gap is noticeable and I KNOW it will affect the results.

Again, the coil are OUTSIDE the magnets, not in between or anything, they are bascially OUTSIDE magnet casing - surrouding it and its the magnet which will be moving, not the coils.

I hope I got it right... cuz if I did, I'm of to paper and pencil and solving it. It good I have all the equations now

But Now i have to do,
use the velocity equation and relate it somehow to the changing area. Because the velocity is vertically up and the for example the velocity equation is y=2x. Y=2x represent the length. dy/dt=dl/dt. So the area is simple dydt*w [width of the rectangle which is OUTSIDE covering the cross section of the wire].

The only confusion I'm facing right now is DO I us the radius of coil of wire to find the total voltage induced? The radius was used before find the B on the square region which was R distance way. And I know the induced is flux over change which I found but shouldn't I be using the circumference somewhere to find the TOTAL induced voltage in the coil instead of find it on the cross section? Maybe * {2piR}? Or it isn't necessary? Like i want total induced voltage.

Last edited: May 15, 2011
21. ### omoplata

315
You can't have the coil outside the magnet for the magnet setup you've chosen. Or at least, use the equation they've given. The equation is valid only for the region between the two magnets. And it is important that you have a 'half ring' of steel connecting the two magnets as shown, because that forms a magnetic circuit, without which the equation will be invalid. Also, as they've stated, it is important that you have enough steel to make sure that it is not saturated (there is enough steel for all of the flux from the outside of the left magnet to be connected to the outside of the right magnet).

So you'll have to use exactly this magnet setup and put your coil between the magnets, or choose another magnet setup. I can't see any way to calculate $$B$$ that will go through the coil for the magnet setup in the diagram you've drawn.

When I started looking at this, I had no knowledge of things like $$B_{r}$$. But since I've read a little now, I think the approximate calculation of flux is possible for a setup where there is a single permanent magnet surrounded by a coil, and the magnet moves up and down, with the knowledge of $$B_{r}$$, and this equation. Once the top of the magnet drops below the coil, the distance to the coil is $$x$$, and that equation can be used. Same thing can be done when the bottom of the magnet is above the coil, only the direction of $$B$$ is reversed. But I don't think this is a good way to maximize flux change, because the change of flux for a change of $$x$$ seems to be small. (I don't have much time to draw images. I will be glad to explain more elaborately if what I'm writing is not clear.)

And looking at their equations, what they've done with steel is assume that steel just extends the thickness of the magnet. So it might be possible to have more magnets with steel pieces one of top of the other (i.e. Steel pieces making contact with North or South poles of magnets) . But let me look in to that first before answering definitely. I have to find out how much steel will give the effect of extending the thickness. I'm sure just adding an infinite length of steel would not make an infinite length of bar magnet from a small bar magnet. It would require an infinite amount of energy to align the magnetic moments (atoms) inside a steel bar of infinite length along one direction, and a small permanent magnet doesn't have an infinite amount of energy.

Any flux region associated with a magnet will move up as the magnet moves up. Well, for the speeds that are important to a project like this, anyway. The flux regoin will have trouble catching up only when the speed of the magnet approaches the speed of light.

I have trouble understanding your second diagram. Could you mark the North and South poles of the magnet?

BTW, could you mark the North and South poles of your first diagram too? Maybe I might have misinterpreted the whole thing.