How to calculate the voltage induced in a coil by an AC magnetic field?

[Psyrebro]

TL;DR Summary

1.Straight conductor has an AC current running through it which makes alternating magnetic field
2.Next to it is a coreless coil

-How to calculate voltage induced at the ends of the coil if it's stationary?
-How to calculate voltage induced at the ends of the coil if it's moving alongside the straigh conductor?

So, as it says in the title, I am trying to calculate overall voltage induced in a coreless coil in the cases of it being stationary and moving in an alternating magnetic field. To go more into detail, I would like to create a mathematical model of a coil in an alternating magnetic field that emenates (for this purpose) from a straight, infinite conductor. The prospect of the coil moving in an already alternating magnetic field makes this calculation seem quite tricky to do (to me at least), and I was unable to find definitive answers both on the internet and in the books.

The questions are:

-How to calculate voltage induced at the ends of the coil if the coil is stationary in the alternating magnetic field of a straight conductor?
-How to calculate the voltage if the field is alternating and the coil is MOVING alongside the straight conductor?


The problems I'm facing are accounting for the very harsh weakening of a magnetic field with distance. Even if the coil has a diameter of couple of cm, there is a big disparity in magnetic field strength between the near and far side of the coil (relative to the straight conductor). Also, I am kind of confused by thinking about Faradays law in AC environment and the interaction with the moving coil.

Sidequestion: If I were to move the coil physically, how would i calculate the physical resistance it generates cause of repelling magnetic fields?

My attempt(EDIT1):
d1=distance from the straight conductor to the near side of the coil
d2=distance from the straight conductor to the far side of the coil
B1=(μ0∗I)/(2∗π∗d1)
B2=(μ0∗I)/(2∗π∗d2)
Bavg=(B1+B2)/2
I would reduce it by 30% because of radial nature of the magnetic field which weakens linearily with distance and the coil which overlaps with the magnetic field so:
Bavg=0,7*(B1+B2)/2
and finally
Ep=A*N*B*2*π*f
Erms=Ep/sqrt(2)

 

[Baluncore]

-How to calculate voltage induced at the ends of the coil if the coil is stationary in the alternating magnetic field of a straight conductor?
-How to calculate the voltage if the field is alternating and the coil is MOVING alongside the straight conductor?

The right hand rule gives you the magnetic field around the straight conductor. You will see that the primary magnetic field does not pass through the turns of your secondary coil. The coil can therefore be replaced with a short straight conductor between the coil terminals. But until you close the loop of the secondary conductor you will have no secondary loop area so cannot compute the voltage.

Since you have only two parallel conductors, the primary being long and the secondary short, you can only calculate the theoretical mutual inductance between those two straight inductors. See "Pair of parallel wires" https://en.wikipedia.org/wiki/Inductance#Self-inductance_of_thin_wire_shapes

 

[Psyrebro]

Should I rotate the secondary coil by 90° relative to the sketch so that coil turns "cut" the magnetic field lines? If so, the calculation doesn't change as much.

For the purpose of this calculation, the load on the secondary coil isn't important, also secondary loop area can be calculated, the leads should be considered short and therefore negligable in the area calculation. In that case, the voltage should indeed appear, but the current isn't important right now as it depends on the load we give it (i will calculate reactance when I am sure that i have the right premise).

Please correct me if I'm wrong!

 

[Baluncore]

Should I rotate the secondary coil by 90° relative to the sketch so that coil turns "cut" the magnetic field lines? If so, the calculation doesn't change as much.

You might draw the secondary coil as a single circular turn on the diagram where the primary conductor is drawn as a straight line. I don't know what you are trying to do, so I really can't say what you should do.

If there is a steady DC current in the primary line then moving the coil parallel to the primary will not change the field cutting the coil, so the voltage should remain zero.
If there is an AC current in the primary line then it will induce an AC voltage in the secondary. Movement of the secondary parallel with the primary will not change the AC voltage induced.

 

[Psyrebro]

If there is an AC current in the primary line then it will induce an AC voltage in the secondary. Movement of the secondary parallel with the primary will not change the AC voltage induced.

Basically, my thoughts are that if we for example take 50Hz AC current, the field switches 50 times in a second and with it, direction of the magnetic field. So if the coil was to be physically moved by some external force along the length of the straight conductor, it would experience being pushed back 25 times in a second because it creates its own magnetic field which is always opposed to the one created by the straight conductor, and also being pushed forward 25 times in a second.

Would those "jitters" cancel out in such a small timeframe resulting in zero or negligable resistance to movement? Or is there something else at play here...like the coil storing magnetic energy and resisting movement?

 

[Baluncore]

1. The original question and title asked for the coil voltage. That suggested the coil is open circuit, so no current flows, and there would be no force due to current in the coil. Now you have changed the game completely by replacing the voltage sensing coil with a shorted turn that will distort the field of the primary conductor.

2. Your numbers are a bit out. 50Hz AC repeats 50 full cycles per second. There are 100 reversals of the field per second. There are 50 positive peaks and 50 negative peaks, making 100 peaks per second.

3. The fields created about the primary and secondary will be in phase and both proportional to primary current. If the primary current is Sin(wt) then the secondary current will be proportional to Sin(wt). The force between the two will be the product of those currents, proportional to Sin²(wt) = 1 + Sin(2wt). That is independent of the current direction, so the “jitter” would have a frequency of 100 Hz. You may have noticed that the sound made by a heavily loaded power transformer is at twice the supply frequency. (One octave higher than the supply).

4. The forces between the primary conductor and the current loop will be perpendicular to the primary conductor. There is no force acting on the secondary coil parallel with the primary conductor, so as you move the coil along the primary you will not feel any longitudinal “jitters”. All magnetic forces will be radial to the primary conductor, at twice the supply frequency.

 

[Psyrebro]

1. The original question and title asked for the coil voltage. That suggested the coil is open circuit, so no current flows, and there would be no force due to current in the coil. Now you have changed the game completely by replacing the voltage sensing coil with a shorted turn that will distort the field of the primary conductor.

I'm still in college, and while I was taught how to calculate standard examples of a rotating coil in a constant magnetic field, or a stationary coil in an alternating magnetic field, shift few things around like in this example and I am lost. I can guess and calculate things but without a real way to test and see for myself, I have to rely on other peoples knowledge and experience and you have my thanks for sharing yours.

2. Your numbers are a bit out. 50Hz AC repeats 50 full cycles per second. There are 100 reversals of the field per second. There are 50 positive peaks and 50 negative peaks, making 100 peaks per second

You are absolutely correct, can't believe I wrote something so dumb.

You may have noticed that the sound made by a heavily loaded power transformer is at twice the supply frequency. (One octave higher than the supply).

I actually haven't noticed it. Gonna keep it in mind when i get the chance.

So what I got from what you wrote is that depending on the orientation of the coil in regards to the straight conductor, it is possible for their magnetic fields to repel each other.
Think of a coil as you said like a circle laying flat just above the straight line conductor. On the sides of the coil, current going through it would actually produce a magnetic field opposite to the one generated by the straight conductor. So that means that there would be a small force applied for every peak, which would result in 100Hz vibration of both the coil and the conductor. Intensity of the vibration would then depend on the current going through the straight conductor.

 

[Baluncore]

You learn the principles quickly because you are not afraid to make mistakes and you express yourself clearly.

Plot the graph of Sin( x ) and of Sin²( x ) over the range 0 to 2*Pi. Then stand outside an electrical substation or under a pole transformer and listen to the hum during a period of high demand. The forces between conductors, and magnetostriction of the core, both produce the distinctive 2'nd harmonic that is an indication or warning of high magnetic flux in a transformer.