The apparatus shown in the photograph above are two types of magnetometer. The unit at left (the small black box connected to the grey box next to it, with the cable going to the probe held in the clamp on the stand) is a Vernier Software magnetic field sensor. The sensing element is a Hall probe (at the bottom of the clear plastic tube held in the clamp above the U magnet), which is connected to an amplifier circuit, whose output (a DC voltage) depends on the magnitude and sign of the voltage generated in the Hall probe. The output of the amplifier is connected to a multimeter, whose readout is shown on the large display.

The unit at right is a Rawson-Lush 720 rotating coil gaussmeter. The motor drive is inside the brown cylinder sitting on the wooden V block, and the rotating coil is near the end of the long 1/4-inch-diameter probe, between the poles of the U magnet. The (DC) voltage generated by the probe is placed across the volmeter at the front right corner of the table, which is calibrated to read in kilogausses.

A note on units follows the explanations of how these two instruments work.

Demonstrations 68.57 – Hall effect, and 68.58 – Hall effect, new, show the potential that develops across the width of a conductor carrying a current along its length, when it is placed in a magnetic field. As the names of these demonstrations would indicate, this phenomenon is called the Hall effect, after Edwin H. Hall, who discovered it in 1879 (American Journal of Mathematics, Vol. 2, No. 3 (Sep., 1879), pp. 287-292). When a conductor sits in a magnetic field, the charges flowing through it are subject to a sideways force, F = qv_d × B, where q is the charge on each charge carrier, v_d is the drift velocity of the charge carriers (given by j/ne, where j is the current density (in amperes/unit area cross-section), n is the number of charge carriers per unit volume, and e is the charge per carrier). The resulting gradient in charge density across the conductor results in the aforementioned potential difference across the conductor, V_xy, where x and y are points opposite each other at the sides of the conductor. This sets up an electric field across the conductor, E_H = V_xy/d, where d is the width of the conductor (or the distance between x and y). This electric field opposes the sideways drift of the charge carriers until a point of equilibrium is reached where it just cancels the sideways magnetic deflecting force, and qE_H + qv_d × B = 0, or E_H = -v_d × B. When the drift velocity and the magnetic field are at right angles, their magnitudes and that of the Hall electric field are related by E_H = v_dB. If we substitute j/ne for v_d, we get E_H = (j/ne)B, or n = jB/eE_H, or n = (jBd)/(eV_xy), which we can rearrange to B = V_xy(ne)/(jd). n, e, j and d depend on the characteristics of the conductor and the current flowing through it. (e is usually one electron charge, but whether it is positive or negative depends on what the charge carriers are.) We see, however, that once these are set, there is a linear relationship (at least over a certain range) between the magnitude of the magnetic field, B, and the voltage developed across the conductor, V_xy; B = kV_xy, where k = (ne)/(jd). Thus, we can design a piece of material that when operated at a particular current, produces a voltage across its width that varies linearly with the magnetic field in which we set it. Such a device, called a Hall probe, allows us to measure the magnitude and direction of a magnetic field whose strength we wish to measure.

The Vernier instrument shown above is made to have an offset of about 2 volts, and has two sensitivity settings. On the “LOW ×10” setting, the sensitivity is 32 gauss per volt, and on the “HIGH ×200” setting it is 1.6 gauss per volt. The offset is difficult to set precisely, and it does not agree exactly between the two settings. (This is a setting that requires opening the unit, and is not meant to be done on a regular basis.) If you wish to make a measurement, then, the best way is to find the maximum and minimum readings, and divide the difference by 2. Multiplying the result by the sensitivity gives the magnetic field in gauss. The probe has a white dot on one face. If the white dot faces the north pole of a magnet, the output voltage decreases. If the opposite side of the probe faces the north pole of the magnet, the output voltage increases. The field between the poles of the U magnet shown in the photograph exceeds the range of this instrument in either setting. The voltage shown on the display is the maximum output voltage. Had the probe been set some distance above the magnet, the reading should have been within the range of the instrument.

On the “HIGH ×200” setting, this instrument gives a measurable response to the earth’s magnetic field. An attempt to measure this in the lecture demonstration preparation area gave a value for the horizontal component of the field (the component to which a compass needle responds) of about 0.19 gauss, and for the total magnetic field (with which a dip needle aligns itself), about 0.63 gauss. Geomagnetic models give 0.24 gauss and 0.46 gauss for these, respectively. (See demonstration 68.09 – Dip needle.) It is quite possible that stray magnetic fields from the building, for example, from rebar in the floor, or from the demonstration tables or other structures in the room, affected these measurements. Still, they are reasonably close to those we might expect based on the models.

In the Rawson-Lush instrument, a coil, whose diameter is 1/8 inch, sits near the end of the probe on a long shaft, which is turned by a motor that is inside the brown cylinder in front of the handle. The way this instrument works is similar to the way a generator works. (See demonstration 72.12 – AC/DC generator.) As the coil rotates in the magnetic field to be measured, the induced EMF is E = -NdΦ_B/dt. N is the number of turns in the coil, and the derivative is the change in magnetic flux with time. The magnetic flux is Φ_B = ∫B · dS, where dS is a surface element, and the integral is over the surface bounded by the coil, or the area of the coil. If the magnetic field is perpendicular to the shaft on which the coil is rotating, the flux varies sinusoidally as the coil rotates, and so does the induced voltage across the coil. A cam-operated breaker mechanism switches the polarity of the coil connections at every half rotation, so that the output voltage resembles a rectified sine wave; all the half cycles are positive. On the back of the motor housing (near the handle) is an arrow, with the ends labeled “N” and “S.” When this arrow is lined up with the magnetic field, the breaker mechanism switches the coil polarity exactly when the generated voltage goes through zero, and the reading is maximum. As you rotate the handle away from this position, the voltage decreases as the cosine of the angle between the arrow and the direction of the magnetic field. The magnetometer reads the component of the magnetic field that is in the direction of the arrow and perpendicular to the probe axis. The letters “N” and “S” indicate the north and south poles of the magnetic field being measured. Assuming that you have these aligned in the direction of the magnetic field, if you reverse the direction of the arrow, the voltage goes negative, which pins the meter against zero. (Which is not good to do!)

The meter has six full-scale ranges, which are 0.4, 1.2, 4, 12, 40 and 120 kilogauss. In the photograph it is set on the 1.2 kilogauss range, and the reading is about 0.75 kilogauss (or 750 gauss).

The brown cylinder near the back of the table is a standard magnet whose strength is 1,000 gauss, ±0.5%, which you can use to check the calibration of the instrument.

The magnetic field, B, is defined in terms of its effect on a moving charge. If a positive charge q is moving through a particular point with velocity v, and it experiences a force F there, then a magnetic field B is present at that point, and satisfies the relation

F = qv × B

The magnitude of this force is

F = qvB sin θ

where θ is the angle between v and B. If we express F in newtons, q in coulombs and v in meters per second, then the units of B are in newtons/[coulomb(meter/second)], or newtons/(ampere·meter). The SI unit of magnetic flux is the weber (Wb). Since the magnetic field B is the flux per unit area, 1 newton/(ampere·meter) also equals 1 weber/meter². The SI name for this unit is the tesla (T). If a magnetic field has a magnetic induction of one tesla, then a charge of one coulomb moving through that field with a component of velocity perpendicular to it of one meter per second, experiences a force equal to one newton.

The CGS unit for magnetic flux is the maxwell (Mx), which equals 10^-8 weber, and the unit of the magnetic field is the maxwell/cm², or gauss (Gs, G). Thus,

1 maxwell/cm² = 10^-4 weber/m² = 10^-4 tesla, and

1 tesla = 1 weber/m² = 10⁴ gauss.

References:

1) NIST Guide to the SI, Chapter 5: Units Outside the SI.
2) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition (New York: John Wiley & Sons, Inc., 1978), pp. 717-721, A32.
3) Sears, Francis Weston and Zemansky, Mark. College Physics, Third Edition (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1960), 607-611.