## FARADAY'S LAW OF ELECTROMAGNETIC INDUCTION: ESTIMATING MAGNETIC FIELD INSIDE OF A MAGNET

If you are interested, visit Introduction to Newtonian Mechanics with Java Applet Simulated Experiments.

### A 'gedanken' experiment

Faraday's law of electromagnetic induction is very abstract for most students taking an introductory physics. A main goal of this experiment is to show how it can be applied for analysis of simple systems. This should make this law more understandable. Its typical textbook formulation states the following: The magnitude of the induced electromotive force (emf) in a loop like circuit equals the time rate of magnetic flux through the circuit. The mathematical equivalent of this statement is given by equation (1)
Equation (2) follows from equation (1) if the later is integrated in the time interval (t1,t2). A general definition of magnetic flux through a loop and its relation to the sign minus in equation (1) are mathematically rather complicated. Fortunately they are not very important on an introductory level. If we deal with a flat loop, like in Fig.1, and magnetic field through this loop is uniform and perpendicular to the loop plane, then the absolute value of the magnetic flux is equal BA. Here B is the magnitude of magnetic field, and A is an area encircled by the loop.

 Fig. 1 Fig. 2 Fig. 3

Let us consider a rod magnet with the cross-section exactly fitting to the loop. Suppose, this magnet moves along the line which is perpendicular to the loop and passes through the loop center. When the magnet is far away, the magnetic flux through the loop is practically equal zero. As the magnet approaches the loop, the flux is gradually increasing. Fig. 2 presents a magnetic field distribution for an 'ideal' rod magnet. All lines of this field inside of the magnet are parallel to the magnet axis. Consequently the magnetic field inside of the magnet is constant. Then, as soon as the magnet enters the loop, the magnetic flux through the loop becomes constant too. In such situation, according to Faraday's law, emf drops to zero. Taking all of it together we shall expect that a voltage V measured along the loop (see Fig. 1) will have a shape which is shown in Fig.3. The time interval T there, shows the time spent by the magnet inside the loop. The initial induced voltage V does not have to be positive. It depends on which magnetic pole enters the loop first, a direction of magnet motion, and how the loop terminals are connected to a measuring device.
What is important there, are two things. First, according to the equation (2) areas, either under positive or negative voltage bumps, represent a maximal magnetic flux through the loop. Second, such maximum is reached when the magnet is inside of the loop. Thus, finding such area from an experimental plot of the induced voltage versus time, and comparing it with the product BA, we can calculate from there a value for the magnetic field B inside of the magnet. This ends our 'gedanken' experiment. Now it is time to make it real.

### A real experiment

There are three unrealistic assumptions in the 'gedanken' experiment. Use of the single loop is not plausible An induced voltage would be too small. Practically, a coil with hundreds of turns must be adopted. For a real magnet its inside magnetic field close to the poles is divergent. Finally, the magnet diameter must be smaller than the coil diameter. The coil must be wound on some kind of pipe, and there should be enough space for the magnet to move through. This is why we can estimate, rather than measure, a strength of magnetic field inside of the magnet.
 Fig. 4 A practical arrangement for this experiment is shown in Fig. 4, which is a vertical cross-section. The pipe is needed there to guide the falling magnet through the coil. The coil terminals must be connected to a device, able to record a single low voltage signal lasting for about 0.1 s. Apple computer or PC running Vernier MPLI (Multi-Purpose Lab Interface) are good choices. One part of MPLI software imitating a low frequency oscilloscope will catch such signal. The another part is capable to integrate a proper part of the signal finding the maximal magnetic flux through the coil. Another choice is a storage oscilloscope with properly set triggering.             Shapes of the recorded signals will be similar to the shape shown in Fig. 3 if the magnet is considerably shorter than the coil. For other cases the time interval T will be zero even for longer magnets. This is related to magnetic field divergency inside of a real magnet. Best for this experiment is to have the magnet and coil of the same length. Once the maximal magnetic flux for the coil is measured we have to find a maximal flux through a single turn of the coil. Thus, a number of turns in the coil must be known. If it is not known additional measurements are needed.             First we have to find the coil wire gauge (or diameter) and material. The material is usually copper. Next, we should measure the coil resistance R. The resistance is related to the material resistivity rw, wire cross-section area Aw, an wire length Lw as follows R = rwLw/Aw. This relation let us find Lw. Finally, measuring an average diameter of the coil turns, we can estimate their number.             If the magnet is not shorter than the coil, then dividing the maximal magnetic flux through the coil by the number of its turns we will obtain a good estimation of the flux through a single turn. This flux is produced by the field inside of the magnet and this part of external magnetic field which still goes through the single turn. The last contribution lowers the flux because the outside magnetic field is in a roughly opposite direction as compared with the inside field. Ignoring this contribution, and dividing the flux through the single turn by the cross-section area of the magnet, we obtain a lower limit estimation for the magnetic field inside of the magnet. Accuracy of this estimation increases as a difference between diameters of the magnet and coil decreases.             If the magnet is shorter than the coil, situation is more complicated. Best estimation for the lower limit of the magnetic field inside of the magnet can be obtained by repeating the same procedure as described above with one difference. Namely, instead of taking a total number of turns in the coil, only a number of turns per the magnet length should be considered.
In our Introductory Physics Lab measurements for three different coils with Brown & Sharpe gauge 30 (diameter 0.0002548 m). In all cases the same AlNiCo magnet with the diameter of 9 mm was used. The summary of results is shown in the table below.

 coil length [m] coil average diameter [m] coil resistance [ohm] estimated number of turns measured maximum flux [Vs] estimated magnetic field [T] 0.055 0.032 19.7 588 0.020 0.35 0.105 0.028 79.5 2722 0.087 0.32 0.127 0.017 14.9 834 0.023 0.26

In Fig. 5 there is the graph of induced voltage (described there as potential) on the last coil from the table above. This graph was recorded with help of MPLI on APPLE IIe computer. The part of the graph between two long vertical lines was integrated to obtain the maximum of magnetic flux through the coil.

Fig. 5

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