MAGNETIC INDUCTION FIELD OF THE EARTH The earth possesses a magnetic field, as anyone who has used a compass knows. Although the existence of the field is no doubt related to the presence of ferromagnetic iron in the earth's core, the mechanism by which the field arises is imperfectly understood. The symmetry of the field is similar to that of a bar magnet or to the field generated by current flowing in a circular coil. the object of this experiment is to measure the magnitude the direction of the earth's field at one station in the laboratory. The device used is called a rotating coil magnetometer and is based on the principle of electromagnetic induction. Theory and Apparatus Definitions By definition, the direction of a magnetic induction field, B, is the direction in which the north pole of a freely-suspended compass needle points when at rest in the field. Thus B is a vector quantity. The magnitude or strength of the field is defined in terms of the force which the field exerts on a charged particle moving in it. This definition, however, is rather awkward to apply in actual measurements of B. In this experiment, B will be obtained from a measurement of the emf induced in a rotating coil. From experiments with magnetic compasses and magnetometers, the large scale symmetry of the earth's field is known to resemble the sketch below in Fig. 1. At some arbitrary location on the earth's surface B has both a horizontal and a vertical component as shown in Fig. 2. The angle which B makes with the horizontal is called the angle of dip. At any point on the equator the field is horizontal (i.e., B v = 0). At the poles it is vertical (i.e., B h = 0).
The Magnetometer and Principles of Operation Magnetic field will be measured with a device called a rotating coil magnetometer. Examine the instrument and identify the two large Helmholtz coils - coils C - and the smaller, centrally-located search coil - coil S. Coil S is rotatable, being attached to a shaft. Fig. 1 Symmetry of earth's Fig. 2 Angle of dip of earth's magnetic field. magnetic field. driven by a small electric motor. The magnetometer you are supplied with is only for student use, having much larger dimensions than the ones manufactured commercially, but the principles are the same. When a coil of wire, say coil S, is placed in a magnetic field of strength B, say the earth's field, an emf can be induced in the coil by changing the flux through the coil. If B is the magnetic flux through the coil, Faraday's law gives the induced emf as d = - B, dt The flux can be changed in a number of different ways. Two are of interest to us: 1. by varying the magnetic induction, B, with time, when the coil is held stationary, and 2. by rotating the coil about an appropriate axis when the field is constant. Method 1 is employed in the experiment, "Magnetic Induction Field of a Circular Coil". Method 2 is used here. 2
Let Fig. 3 on the next page represent coil S. Suppose B is perpendicular to the axis of rotation, and at an angle with respect to the normal to the plane of the coil. Then by definition, the magnetic flux through the coil is given by the area of the coil and B is the absolute value of the field strength. For a coil with n turns of wire, B = nba cos, Fig. 3 The Rotating Coil. If the motor rotates the coil at an angular velocity/frequency of ω rad sec. -1, then θ = ωt and from eqs (1) and (2), (t) = nab sin t = o sin t This emf, appearing across the terminals labelled `rotating coil' on the magnetometer, can be measured. It is large enough not to require amplification. Note that the apparatus can be laid flat or tipped up on end. This enables the search coil to be rotated about either a horizontal or a vertical axis - and hence sample either B v or B h. Peak-to-Peak and RMS Detection For the most part, the emf will be measured with the multimeter. But an oscilloscope can also be used. When the oscilloscope is used, peak-to-peak (pp) values are obtained as shown below in Fig. 4. The multimeter, however, displays a root-mean-square (rms) value. This is because when an AC voltage is applied to the multimeter set to the AC volts mode the signal is first rectified and then filtered. The result is a DC voltage equal to the root-mean-square of the AC input. Fig. 4 Relationship between sinusoidal signal and its DC root-mean-square. 3
That is, and rms 1 = 2 2 pp pp = 2 o = 2 2 rms, The Oscilloscope - An Introduction Fig. 4 shows a front view of the Philips model PM3232 oscilloscope used in this experiment. It is a dual beam device, but you will use only one beam - beam B. As can be seen from the figure there are too many controls for each one of them to be discussed separately here. Fig. 5 Front view of Philips oscilloscope The following familiarization routine is intended only for those students who have not used a scope before and is designed to yield a Lissajous figure - a necessary condition for measuring frequency in this experiment. Before switching on, carry out the following checks: 1. intensity control mid-range, 4
2. A-input and B-input lever switches in AC position, 3. A- and B-amplification controls fully counter-clockwise, variable controls in `cal' position, 4. A and B beam selectors `in'. 5. Time base control fully counter-clockwise, in `via Y A ' position, 6. X-position and Y B -position midrange. Now switch on the scope and allow 15 sec. for warm up. 7. Apply the signal to be studied to the B-input and the standard signal (from the signal generator) to the A-input. Note that the A-input signal provides the time base. 8. Rotate the A- and B-amplification controls clockwise until you observe a figure of useful size. 9. Finally, vary the frequency of the standard signal until you obtain a Lissajous figure which is recognizable. Measuring Frequency In this experiment the frequency of a signal is measured using the technique of Lissajous figures. The signal whose frequency is to be measured is applied to the Y B -input of the scope and the output of a signal generator is applied to the Y A -input (switched to time base). The two signals thus superimposed at right angles produce a Lissajous figure. Fig. 6 shows two useful ones. Obviously, if a circle is obtained, the signals are of the same frequency and the value can be read off the dial of the signal generator. The Experiment Assembly and Trial Run Begin by assembling the apparatus. Plug the motor which rotates coil S into the variac. Connect the output of coil S - `Rotating Coil' - to the B-input of the scope and put a Philips multimeter in parallel with the input. Connect the HP model 200AB signal generator to the A-input 5
of the scope. Get the coil spinning at a modest rate and vary the frequency of the generator until you obtain a Lissajous figure. if the controls on the scope are confusing, try following the familiarization routine in the previous section. The Lissajous figure is not always stationary. Why? The Field to be Measured Your job is to measure the magnitude and direction of the earth's magnetic field where your apparatus happens to be located. To get a feeling for the direction of the field use a compass and dip needle. What is the angle of dip? Note carefully the direction of B h. There are two methods by which the magnetometer can be used to measure the earth's field. One is a direct method, the other, a null method. Direct Method Use your knowledge of the direction of B h to orient the apparatus so as to sample B h and B v separately. How must the apparatus be oriented so that only one component is sampled at a time? Make sure to measure ε and ω (=2π f) simultaneously. Null Method Null methods generally yield more precise results than direct methods. In this null method a DC current (from the U. of T. power supply) is passed through the pair of Helmholtz coils to produce a magnetic field in opposition to the component of the earth's field under study. Once the two fields are equal and opposite the output ε should read zero for any value of ω. The component being measured must then equal the field produced by the coils at their geometrical centre, given by 8 NI B = o 3/2 (5) R where o = 4π x 10-7 Weber Ampere -1 meter -1 N = number of turns in a single coil I = current in the coils R = radius of the coils Measurements of B h and B v at Agincourt, Ontario, have yielded 0.154 and 0.563 x 10-4 Tesla, respectively (Clark's Tables, 1965). SQ78 6