Help me understand Biot-Savart law
As a student I?m having difficulties understanding the Biot-Savart law?
The problem originates in basic definition of magnetic induction B and its relation with magnetization M and magnetic field H. The induction B in ferromagnetic materials can be written as:
B=n0*(H+M)
where H is the filed that is not affected by the presence of iron material and M is the material contribution to the field? The above equation can be rewritten in well known expression
M=x*H; (1+x)=nr ---> B=n0*nr*H
So now the question ? In above equations, H is the filed that would be produced, for instance, by the current in the vacuum, right? So, H is not affected by the presence of the iron? Thus, this filed can be calculated using Biot-Savart?. We can then calculate induction B, by simply multiplying this value with the material constants?
This is of course not correct, since this way all the geometry (flux path) is neglected. For example, if we would like to calculate the B in a tooth of the electric motor, then we would simple calculate H using Biot-Savart expression and multiply this value with n0*nr?. If above presumption would be correct, the FEM would not be needed at all?.
Can somebody please point me where in above thinking I made a mistake?
Thank you in advance,
Vincenc
Hello Vincenc,
If I understood your question correctly then what you want 2 know is:
1. Why is the conducting structure neglected while using Biot-Savart law ?
2. Is there a difference between Iron, Aluminum and Copper while calculating the magnetic field/flux they create while conducting currents?
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1. Well, first of all most text books use the linear approximation for line current sources. There exists a more accurate expression for H-field due to a volume current distribution (Regarding 3D structures) :
H=(1/4π)*∫((J×R)/R2)*dv
2. If you consider the magnetic susceptibility values (x) for most dielectrics and metals (such as Copper and Aluminum), you'll find that it is ~10e-5 so it can be neglected and then n~n0=1.
Considering ferromagnetic materials as Iron, M field is induced in the material in response to an externally applied H field.
In my opinion you can calculate H-field from a volume current distribution using the above formula (H,n~1), and then calc. M=x*H (x>>1),
and finally B=n0*(H+M).
I hope this help,
Good luck,
P.
a similar question can be raised for the electric field (E) and the electric displacement (D). I think your question is related to the macroscopic definition of electromagnetic fields. If you have the book by J. D. Jackson, chapter 4 may be very helpfull in understanding these issues.
Hi,
Thank you both for your answers!
Indeed, I think that these problems in understanding are because of the difference between macroscopic and differential point of view? I?ll try to find the book you recommended. Thanks again!
Have a nice day!