![]() This is apparent from the cross product between r and d l, and the rotational symmetry of the problem. An analysis of the Biot-Savart law shows that along the z -axis the following is true: while there are transverse component contributions in d B from individual segments d l (pointing in the x - y plane), their sum vanishes due to symmetry (they all cancel each other). We place the loop in the x - y plane with the center at the origin. Calculate the strength of the magnetic field at a distance d away over the center of the loop. We consider now the following problem: a charge-carrying circular loop with radius R produces a magnetic field perpendicular to the plane of the loop. How does your result compare to the Earth's magnetic field at the surface (about 0.04 mT)? Insert numerical values for the radius (in the range of 0.1 m) and the current (in the several A range). Calculate the magnetic field at the center of the loop. How different are the results?Ĭonsider a current-carrying circular loop with radius R. Use quantitative numerical examples to investigate the difference beween both expressions: calculate the field strength at distances away from the wire that are fractions (multiples) of the wire length. In the limit of an infinitely long wire the result previously calculated from Ampere's law is obtained: We orient the wire along the x axis through the origin with - L /2 r:=sqrt(x^2+D^2) ![]() We calculate the magnetic field due to a current passing through a finite straight wire of length L at a distance D away from the middle. Our interest is to make practical use of the Biot-Savart law. Note, however that the direction of the E field is radial, while the B field is orthogonal to the line segment d l as well as to the displacement vector r (as evident from the cross product). This is analogous to the electric field generated by a charged rod. Note the 1/ r ^2 dependence of this expression, which upon integration leads to a 1/ r dependence of the field strength B. One can motivate the Biot-Savart law by performing the direct calculation for a straight-line wire, and then realize that the generalization for the magnetic field contribution at displacement vector r from the line segment d l reads asĭ B = mu/4 Pi I*crossprod(d l, r )/r^3 One can ask the question how the macroscopic magnetic field generated by a current passing through a wire is composed of infinitesimal contributions d B generated by oriented infinitesimal line segments d l. If the Coulomb law for the electric field of a point charge represents the underlying basis for Gauss' law of electrostatics, the Biot-Savart law plays the same role with respect to Ampere's law. The Biot-Savart law serves to calculate magnetic field for situations where Ampere's law is not useful due to a lack of symmetry.
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