# EMF induced in an off-axis solenoid due to a rotating magnet

• Cactus
In summary, the formula used for the induced voltage in a previous experiment is only an approximation and works for z >> R, where R is the coil radius. To get a more accurate result, it is suggested to move the coil away from the z-axis and turn it as it is moved so that its plane is always perpendicular to the radius from the center of the magnet to the center of the coil. By plotting the induced rms voltage as a function of spherical angle theta from the z-axis, the dipolar approximation for the magnetic field predicts that the voltage as a function of angle at fixed radius r varies according to V(theta) = V(0)cos(theta), where V(0) is the rms voltage at zero angle. If the
Cactus
Homework Statement
Hey, I have come up with an idea for my EEI to measure the emf induced in a solenoid by a rotating magnet when the solenoid is not directly above the magnet (such that the magnetic field is not uniform). Previously I had to perform an experiment where the solenoid was on axis (see picture) and now I plan to further the experiment by moving it up and down (increasing height from the magnet) and left and right (increasing horizontal distance from the magnet) and I was wondering if anyone could point me in the right direction for deriving a formula for the off axis emf. Currently below are the two formulas for the on axis emf.

In the experiment it will be possible to measure the emf and thus the V and then calculate the field strength at that point (As N, w and A are constant). This field strength can then be used to determine the magnetic dipole moment, however it could also be used to check what factors affect the strength. So far I'm guessing the off axis magnet will only be affected by distance from the magnet, and thus only the z factor in the final formula will affect the field strength and this could be tested by measuring the dipole and using theoretical z values to find field strength, and comparing this to experimental values of field strength (However I cant check this until I perform the experiment this week).

But apart from that I could imagine the angle between the solenoid and the magnet having an effect, however I can't see how I would derive a theoretical formula for it at this point in time

Any help would be appreciated
Thanks!
Relevant Equations
emf = -wNBAsin(wt)
Where emf is induced in solenoid
w is angular frequency
N is number of turns in the solenoid
B is the magnetic field strength
A is the area of the solenoid

V = wNBA
V is the voltage in the solenoid
w, N, B, A as before

B = u/2pi * 2*U / z^3
B is field strength
u is constant 1.26*10^-6
z is the distance from the centre of the magnet along the axis
U is the magnetic dipole moment

The red arrows correspond to the areas the solenoid will be moved too

Delta2
The formula you used for the induced voltage in your previous experiment is only an approximation of the actual result. It was obtained on the assumption that the B field due to the permanent magnet is uniform over the surface of the pick-up coil (I prefer that name to solenoid) and points in the ##+z##-direction, perpendicular to the plane of the coil. This assumption works for ##z >>R## where ##R## is the coil radius. If you move the coil away from the ##z##-axis, I suggest that you not move it parallel to itself because the dependence of the B-field on the radial direction becomes unnecessarily complicated. Instead, consider turning the coil as you move it so that its plane is always perpendicular to the radius ##r## from the center of the magnet to the center of the coil. Then plot the induced rms voltage as a function of spherical angle ##\theta## from the ##z##-axis. The coil must be in the ##xz## plane. In that case, the dipolar approximation for the magnetic field that you have already used predicts that the voltage as a function of angle at fixed radius ##r## varies according to ##V(\theta)=V(0)\cos\theta##, where ##V(0)## is the rms voltage at zero angle that you have already investigated.

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kuruman said:
[...] ##V(\theta)=V(0)\cos\theta##, where ##V(0)## is the rms voltage at zero angle that you have already investigated.
Not to be nitpicky, but I think you mean "peak" voltage (rather than rms voltage).

kuruman
collinsmark said:
Not to be nitpicky, but I think you mean "peak" voltage (rather than rms voltage).
Thanks. Message edited to convey that ##V(0)## is whatever is measured at zero angle, peak or rms. I had rms because that's what one usually measures with a voltmeter.

collinsmark
kuruman said:
The formula you used for the induced voltage in your previous experiment is only an approximation of the actual result. It was obtained on the assumption that the B field due to the permanent magnet is uniform over the surface of the pick-up coil (I prefer that name to solenoid) and points in the ##+z##-direction, perpendicular to the plane of the coil. This assumption works for ##z >>R## where ##R## is the coil radius. If you move the coil away from the ##z##-axis, I suggest that you not move it parallel to itself because the dependence of the B-field on the radial direction becomes unnecessarily complicated. Instead, consider turning the coil as you move it so that its plane is always perpendicular to the radius ##r## from the center of the magnet to the center of the coil. Then plot the induced rms voltage as a function of spherical angle ##\theta## from the ##z##-axis. The coil must be in the ##xz## plane. In that case, the dipolar approximation for the magnetic field that you have already used predicts that the voltage as a function of angle at fixed radius ##r## varies according to ##V(\theta)=V(0)\cos\theta##, where ##V(0)## is the rms voltage at zero angle that you have already investigated.
Hey, thanks for the reply and insight
I was just wondering when you explaining you're above statement would you mean something like this

where say for the top example if the magnet housing was 5 x 5cm (From memory I think it is) and I wanted to measure at a distance r = 7.5cm, I would move the magnet around keeping it perpendicular with the dipole moment of the magnet and keeping it a distance of 7.5cm, so that for an angle of 45 from the x (also then 45 from the z) it would be 7.5sin45 and 7.5cos45 on the x and y coordinates respectively. And then say for the second sketch at the bottom it would be some combination like that, where the angle in your formula you've given corresponds to that angle 30, 45, 60 and 90 from the z (Where the expected value for 90 would be zero, but because the field is not uniform it will be slightly higher experimentally)?

Also, in the case that I'm told I can't perform this experiment, for the original experiment (where the magnet is moved parallel along the x axis) would the calculation/formula be something derivable by hand or would it just be too complicated for that?
Thanks again!

Here is what I had in mind.

The dipole moment rotates in the yz-plane and the sampling circle is in the xz-plane, also the plane of the screen. Within this model, when ##\theta=90^o## the flux through the coil is zero. If you are told that you cannot do it this way, argue that this setup makes sense. The magnetic field depends only on ##r## and ##\theta## and is symmetric about the ##z##-axis. You have already established the dependence on distance when ##r=z##; it makes sense to investigated the angular dependence by changing only the angle and keeping ##r## constant. Furthermore by turning the coil as shown in the figure, you maximize the magnetic flux through it. Moving the coil along the x-axis changes both ##r## and ##\theta## at the same time which makes it harder to sort out which change is responsible for changing the voltage. The idea behind experimental design is to keep it simple, simple to perform and simple to analyze.

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## 1. How does a rotating magnet induce EMF in an off-axis solenoid?

When a magnet is rotated near an off-axis solenoid, the changing magnetic field of the magnet will create a changing magnetic flux through the solenoid. This changing flux will induce an electromotive force (EMF) in the solenoid, according to Faraday's law of induction.

## 2. What factors affect the magnitude of the induced EMF in an off-axis solenoid?

The magnitude of the induced EMF in an off-axis solenoid is affected by the strength of the rotating magnet, the distance between the magnet and the solenoid, the speed of rotation, and the orientation of the solenoid relative to the magnet's axis of rotation.

## 3. How is the direction of the induced EMF determined in an off-axis solenoid?

The direction of the induced EMF in an off-axis solenoid is determined by Lenz's law, which states that the induced current will flow in a direction that opposes the change in magnetic flux through the solenoid. This means that the direction of the induced EMF will be such that it creates a magnetic field that opposes the changing magnetic field of the rotating magnet.

## 4. Can the induced EMF in an off-axis solenoid be increased?

Yes, the induced EMF in an off-axis solenoid can be increased by increasing the strength of the rotating magnet, decreasing the distance between the magnet and the solenoid, increasing the speed of rotation, or changing the orientation of the solenoid relative to the magnet's axis of rotation.

## 5. How is the induced EMF in an off-axis solenoid used in real-world applications?

The induced EMF in an off-axis solenoid is commonly used in generators and motors, where the rotating magnetic field of a magnet is used to produce electrical energy or convert electrical energy into mechanical energy. It is also used in sensors and detectors that measure changes in magnetic fields, such as in magnetic resonance imaging (MRI) machines.

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