LORENTZ COMPASS 3.0.10.9 62
Hullen Vilius
To avoid "same direction" answers, let's assume the B and H are pointing in different direction, i.e., the magnetic permeability is not a diagonal matrix. What direction will a compass (theoretically) embedded inside a magnetic medium points to?
I feel B-field is the answer because the Lorentz force law f=q(E+v*B) involves only E-field and B-field, regardless of whether the medium is magnetic or dielectric. And since the operation of the compass must involve Lorentz force in some way, it can therefore only depends on B-field and E-field?
Usually treatment of H and B is superficial. While there is a chapter in Jackson which derives the macroscopic fields from the microscopic, Jackson also refers to Robinson's 'Macroscopic Electrodynamics' for further information.
An average which seems more useful than either the above-defined h or b is the line average of the microfields perpendicular to the magnetization, since this will determine the magnetic deflection of a charged subatomic particle moving rapidly through the medium. I have not found a rigorous argument to prove that this is equal to b, and there are good reasons for supposing that it will not be b, in general, for an ordered medium. However, if the medium is random, thereseem good informal reasons to suppose it will be equal to b over a finite path, since the series of microscopic paths described effectively covers all field possibilities. This appears to have been confirmed experimentally [Rasetti].
The deflection of mesons in a magnetized ferromagnetic medium was investigated. A beam of mesons was made to pass through 9 cm of iron, and the resulting distribution of the beam was observed. Two arrangements were employed. In the first arrangement, the deflection due to the field caused a fraction of the mesons to hit a counter placed out of line with the others. An increase of sixty percent in the number of coincidences was recorded when the iron was magnetized. In the second arrangement, all the counters were arranged in line, and the deflection due to the field caused an eight percent decrease in the number of coincidences. These results are compared with theoretical predictions deduced from the known momentum spectrum of the mesons and from the geometry of the arrangement. The observed effects agree as wellas can be expected with those calculated under the assumptions that the effective vector inside the ferromagnetic medium is the induction B, and that the number of low energy mesons is correctly given by the range-momentum relation.
A MEMS magnetic field sensor is a small-scale microelectromechanical systems (MEMS) device for detecting and measuring magnetic fields (Magnetometer). Many of these operate by detecting effects of the Lorentz force: a change in voltage or resonant frequency may be measured electronically, or a mechanical displacement may be measured optically. Compensation for temperature effects is necessary. Its use as a miniaturized compass may be one such simple example application.
Magnetometers can be categorized into four general types[1] depending on the magnitude of the measured field. If the targeted B-field is larger than the earth magnetic field (maximum value around 60 μT), the sensor does not need to be very sensitive. To measure the earth field larger than the geomagnetic noise(around 0.1 nT), better sensors are required. For the application of magnetic anomaly detection, sensors at different locations have to be used to cancel the spatial-correlated noise in order to achieve a better spatial resolution. To measure the field below the geomagnetic noise, much more sensitive magnetic field sensors have to be employed. These sensors are mainly used in medical and biomedical applications, such as MRI and molecule tagging.
There are many approaches for magnetic sensing, including Hall effect sensor, magneto-diode, magneto-transistor, AMR magnetometer, GMR magnetometer, magnetic tunnel junction magnetometer, magneto-optical sensor, Lorentz force based MEMS sensor, Electron Tunneling based MEMS sensor, MEMS compass, Nuclear precession magnetic field sensor, optically pumped magnetic field sensor, fluxgate magnetometer, search coil magnetic field sensor and SQUID magnetometer.
Quality factor is a measure of how much energy can be maintained during vibration of the resonator. There might be several factors that can damp the resonator, such as mechanical damping of resonator itself or damping from outside pressure and temperature. [2]
Resonance frequency is the frequency at which the device vibrates with the highest amplitude (or the longest, as a struck bell or tuning fork). Resonance frequency is governed by geometry of the device. We can calculate resonance frequency when we know dimension of the device, equivalent Young's modulus of the device, and the equivalent density of the device. [3]
Responsivity (which contributes to resolution) describes the size of the oscillation we can get from devices with same external condition. If we apply the same current and B field to several resonators, devices that show larger vibration amplitudes are said to have a higher responsivity. All other things being equal, a higher responsivity device is more sensitive. The range of magnetometers based on piezoelectric resonators is mV/T (millivolt/Tesla), so higher responsivity is generally better.[5]
Resolution refers to the smallest magnetic field a device can measure. The smaller the number, the more sensitive the device. The range of magnetometers based on piezoelectric resonator is a few nT (nanoTesla).[5]
A MEMS-based magnetic field sensor is small, so it can be placed close to the measurement location and thereby achieve higher spatial resolution than other magnetic field sensors. Additionally, constructing a MEMS magnetic field sensor does not require the microfabrication of magnetic material. Therefore, the cost of the sensor can be greatly reduced. Integration of MEMS sensor and microelectronics can further reduce the size of the entire magnetic field sensing system.
This type of sensor relies on the mechanical motion of the MEMS structure due to the Lorentz force acting on the current-carrying conductor in the magnetic field. The mechanical motion of the micro-structure is sensed either electronically or optically. The mechanical structure is often driven to its resonance in order to obtain the maximum output signal. Piezoresistive and electrostatic transduction methods can be used in the electronic detection. Displacement measurement with laser source or LED source can also be used in the optical detection. Several sensors will be discussed in the following subsections in terms of different output for the sensor.
Beroulle et al.[6] have fabricated a U-shape cantilever beam on a silicon substrate. Two piezo-resistors are laid on the support ends. There is an 80-turn Al coil passing current along the U-shape beam. A Wheatstone bridge is formed by connecting the two "active" resistors with another two "passive" resistors, which are free of strain. When there is an external magnetic field applied to the current carrying conductor, motion of the U-shape beam will induce strain in the two "active" piezo-resistors and thereby generate an output voltage across the Wheatstone bridge which is proportional to the magnetic field flux density. The reported sensitivity for this sensor is 530 m Vrms/T with a resolution 2 μT. Note that the frequency of the exciting current is set to be equal to the resonant frequency of the U-shape beam in order to maximize the sensitivity.
Herrera-May et al.[7] fabricate a sensor with similar piezoresistive read-out approach but with different mechanical motion. Their sensor relies on the torsional motion of a micro-plate fabricated from silicon substrate. The exciting current loop contains 8 turns of aluminum coil. The location of the current loop enables a more uniform Lorentz force distribution compared with the aforementioned U-shape cantilever beam. The reported sensitivity is 403 mVrms/T with a resolution 143 nT.
Kdr et al.[8] also chose the micro-torsional beam as the mechanical structure. Their read-out approach is different. Instead of using piezoresistive transduction, their sensor relies on electrostatic transduction. They patterned several electrodes on the surface of the micro-plate and another external glass wafer. The glass wafer is then bonded with the silicon substrate to form a variable capacitor array. Lorentz force generated by the external magnetic field results in the change of capacitor array. The reported sensitivity is 500 Vrms/T with a resolution of a few mT. The resolution can reach 1 nT with vacuum operation.
Emmerich et al.[9] fabricated the variable capacitor array on a single silicon substrate with comb-figure structure. The reported sensitivity is 820 Vrms/T with a resolution 200 nT at the pressure level of 1mbar.
Sunier et al.[10] change the structure of aforementioned U-shape cantilever beam by adding a curved-in support. The piezoresistive sensing bridge is laid between two heating actuation resistors. Frequency response of the output voltage of the sensing bridge is measured to determine the resonant frequency of the structure. Note that in this sensor, the current flowing through the aluminum coil is DC. The mechanical structure is actually driven by the heating resistor at its resonance. Lorentz force applying at the U-shape beam will change the resonant frequency of the beam and thereby change the frequency response of the output voltage. The reported sensitivity is 60 kHz/T with a resolution of 1 μT.
Bahreyni et al.[11] fabricated a comb figure structure on top of the silicon substrate. The center shuttle are connected to two clamped-clamped conductors used to change the internal stress of the moving structure when external magnetic field is applied. This will induce the change of the resonant frequency of the comb finger structure. This sensor use electrostatic transduction to measure the output signal. The reported sensitivity is improved to 69.6 Hz/T thanks to the high mechanical quality factor (Q = 15000 @ 2 Pa) structure in the vacuum environment. The reported resolution is 217 nT.
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