By Jerrold Franklin
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Extra resources for Classical electromagnetism
The contribution to the magnetic ﬁeld at a distance r from the origin of the coordinate system that is due to a differential length of the current dz which is at a distance R from the point is dB = μ0 Idz sin θ 4πR2 The direction of this B ﬁeld is, according to the Biot–Savart law, perpendicular to the plane containing the positive z axis and the unit vector from the current element I dz to the point according to the right-hand rule: az × aR . Hence, it is directed circumferentially about the current.
13) where aφ = az × ar . 14) Determination of the direction of the magnetic ﬁeld of a current is obtained with the famous right-hand rule. ] According to the Biot–Savart law, if we place the thumb of our right hand in the direction of the current I, the ﬁngers of that hand will give the resulting direction of the B ﬁeld, which is perpendicular to the plane containing (1) the current I and (2) the vector pointing from the current to the point at which we desire to determine the B ﬁeld; aφ = az × ar .
Bx (y, z), By (x, z), and Bz (x, y)], the divergence of B will automatically be zero. But there are obviously many cases where the vector components of B are functions of some or all of the axis variables x, y, and z, yet the divergence of B is still zero. 11) The dc current density vector is denoted as J, whose units are A/m2 , and v is the volume containing this current. A differential segment or “chunk” of this current density vector contains J dv ampere-meters, and the distance from this chunk of current (the source of the B ﬁeld) to the point at which we are computing the magnetic ﬁeld B is denoted as R.
Classical electromagnetism by Jerrold Franklin