Lab 6.
Electromagnetic Induction.
Theoretical Models.

Model 1.


Figure M.1
Figure M.2
In this lab you will measure the flux through the circular loop of the magnetic field of the small magnet located in the center of the loop (see Figure M.1). You will then use the measured flux to determine (again) the magnetic dipole moment of the magnet. To measure this flux, you will use Faraday’s law.

Assume a small magnet moves along a line perpendicular to the plane of the loop of radius  (Figure M.2). The flux  of the magnetic field of the dipole through the loop depends on the distance  between the dipole and the plane of the loop.

Figure M.3
How does the function  look like? It is sufficient to visualize the magnetic field lines of the magnetic dipole (see the previous Lab) to conclude that for the magnet moving along any line perpendicular to the plane of the loop:
  1. Magnetic flux through the loop is the same when the magnet is at the points that are symmetric with respect to the plane of the loop: . Therefore  is an even function.
  2. Magnetic flux through the loop is maximum when the magnet crosses the plane of the loop: .
  3. Magnetic flux vanishes at large distance from the loop:  as .

Figure M.3 depicts a function that has these properties.

The flux of the magnetic field of the dipole moving along the axis of symmetry of the loop is given by (I.11) which we obtained by integrating the vector potential over the loop:


Figure M.4

 

(M.1)

It is easy to see that (M.1) satisfies all aforementioned properties and is of the shape shown in Figure M.3.


Our goal is to find the maximum value of the flux  when the magnet passes through the center of the loop. As the magnet moves along the line perpendicular to the plane of the loop the induced emf in the loop as a function of time is (we do not watch for the signs here):

 

(M.2)

where  is the speed of the magnet. The derivative of the function of the shape of Figure M.3 is shown in Figure M.4.

Figure M.5 Induced emf in the loop as a function of time
  Since  has maximum at  its derivative   at .If the sped of the magnet stays approximately constant as it moves through the vicinity of the loop the shape of the  function mimics that of  as follows from Faraday’s Law (M.2) (see figure M.5). Let us denote the time when the magnet crosses the plane of the loop . From (M.2) it follows that Integrating emf (M.2) with respect to time from   to  yields :


 

(M.3)

Therefore the flux through the circular loop of the magnetic field of the small magnet in the center of the loop is the integral of the induced emf in the loop with respect to time as the magnet moves from a very distant point to the center of the loop along the loop’s axis – this is the area under the  curve from negative infinity up to the time  where  crosses zero ( shown in green in Figure M.5). We may also integrate  from  to :

 

 

(M.4)

The values for  obtained from (M.3) and (M.4) should, in theory, be the same. Unfortunately, iOLab device has small bias for such measurements on High Gain terminal. This means that the  curve is shifted a little – ether up or down. Because of this shift the values for  obtained through integration (M.3) from   to  ( in Figure M.5) and through integration (M.4) from  to  ( in Figure M.5) are slightly different. To exclude this bias, we will take the average of these flux values:

 

 

(M.5)

For the magnet moving along loop’s axis of symmetry equating obtained through (M.5) with the value for  from (M.1) allows to evaluate the magnetic dipole moment . You may compare it with the values you obtained using other techniques in Lab 5. I believe that the value for  obtained in this Experiment is more accurate (closer to the true value) than the values you calculated in Lab 5.

 

 

Model 2.

Figure M.6
In this model you will derive (yet another!) alternative technique for evaluation of the magnetic dipole moment of the magnet from the induced emf  signal as the magnet moves along the axis of the loop (see Figure M.2). In this model we will not use integration to calculate the flux but instead use the value for the induced emf amplitude  (see Figure M.6). I will only provide general guidance here and you will derive the expression that will be used to evaluate magnetic moment from the experimental data.

Our goal is to express the magnetic dipole moment  in terms of emf amplitude .

  1. At first, we need to find  values where  is maximum ( and  in Figure M.4). From symmetry it is clear that .
    1. Differentiate (M.1) with respect to  to find the expression for .
    2. Differentiate  with respect to  to find .
    3. Solve equation  to find  and  where   attains its maximum/minimum values. Express  and  in terms of loop radius .
  2. Calculate . It is clear that .
  3. From Faraday’s Law (M.2), it follows that

     

    (M.6)

    We will assume the magnet moves with constant speed over the interval  (note: the time interval  is very small). Therefore, we may approximate the speed of the magnet as . Express  in terms of loop radius  and plug the speed of the magnet  and your expression for in (M.6).

  4. Express magnetic dipole moment  from the resulting expression – you should have  expressed in terms of , , , and .
  5. To exclude small iOLab’s bias use ( in place of  when calculating  from your experimental data (see Figure M.6)