zac_physics_student said:
Hey guys, recently I have conducted an experiment investigating the effect of temperature (K) against magnetism (mT). I have found two formulas and understand that they are for both linear and non linear relationships but why does the function change from linear?
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Your question of the linearity is quite interesting. In general, the ## M ## in the material responds to the ## B ## in the material, most often in a linear fashion, but in ferromagnetic materials it can reach a saturation point. For linear materials, and really in general, we can always write ## M=\chi' B ## for some ## \chi' ##, where ## \chi' ## may have some ## B ## dependence for large ## B ##. Since we can also write ## B=\mu_o H+M ## , substituting in ## B=M/\chi' ## gives ## M=\mu_o H \frac{\chi'}{1-\chi' } ##. ## \\ ## In many materials, ## \chi '<< 1 ## so that it is straightforward to write ## M=\chi \, \mu_o H ## where ## \chi=\frac{\chi'}{1-\chi'} ##. ## \\ ## When ## \chi' > 1 ##, as it can be in the case of a ferromagnetic material, ## \chi ## becomes negative, and although we can still write ## M=\chi \, \mu_o H ## for this number ## \chi ##, the number ## \chi ## needs to be found from a hysteresis graph of ## M ## vs. ## H ##. In the case of a permanent magnet, the operating point is in the first or 4th quadrant of the hysteresis curve where ## M ## is opposite the direction of ## H ## and ## \chi ## is negative. Notice the constant ## \chi' ## is actually much better behaved than ## \chi ## in that ## M ## and ## B ## will always point in the same direction for paramagnetic and ferromagnetic materials. (Note: For paramagnetic and ferromagnetic materials ## \chi'>0 ##. For diamagnetic materials, ## \chi'<0 ## ). ## \\ ## For additional details, see also the following Insights article:
https://www.physicsforums.com/insights/permanent-magnets-ferromagnetism-magnetic-surface-currents/ (c.g.s. units are used in the article, but the results can be readily converted to M.K.S.). ## \\ ## One additional question you may have is, why the odd behavior for the response of the ## M ## to ## H ##? The answer is ## H ## is not in general a magnetic field. Although it can be partly composed of magnetic fields, in the material the ## H ## from its own poles represents a correction term to the ## B ## from the magnetic surface currents that differs from ## M ## for geometries other than a long cylinder.(And that explains why ## \chi ## is negative in the permanent magnet. This ## H ## in the material from its own magnetic poles (that occur as a result of the magnetization ## M ## ) is simply a correction term to the magnetic field ## B ## which forr a long cylindical geometry has ## B=M ## that arises from the magnetic surface currents from the magnetization ## M ##.) ## \\ ## The magnetic field inside a permanent magnet is ## B=M ## along with the correction term ## \mu_o H ## computed from its own poles which points in the opposite direction. This is one of a couple of reasons why we have the equation ## B=\mu_o H+M ##. In addition to ## H ## from its own poles, ## H ## also consists of contributions from magnetic poles external to the material, as well as from any currents from conductors where it is computed via Biot-Savart's law. ## \\ ## In the case of a sample of material inside a current-carry solenoid, the applied ## B_a=\mu_o H_a ## where ## H_a ## is from the solenoid, but this scenario where this ## H_a ## is treated as the magnetic field ignores the internal ## B=M ## from the magnetic surface currents that result from the magnetization of the material. The applied field is ## B_a=\mu_o H_a ##, and this is what the hysteresis curve ## M ## vs. ## H_a ## uses, but the actual magnetic field in the material is ## B=\mu_o H_a+M ##. (Besides the hysteresis curve of ## M ## vs. ## H_a ##, it could also be useful at times to have a graph of ## M ## vs. ## B ##). ## \\ ## In summary, ## B ## is the actual magnetic field, and ## H ## is actually something of a mathematical construction.
