First write B(r) as a Taylor expansion about the center of the loop where r 0 is the position of the dipole and 0 denotes differentiation with respect to r 0 Put this into the Lorentz force law (Eq 5 16) to obtain Or, numbering the Cartesian coordinates from 1 to 3 where i j k is the Levi Civita sy...

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First write B(r) as a Taylor expansion about the center of the loop: where r 0 is

Question:

First write B(r) as a Taylor expansion about the center of the loop: where r₀ is the position of the dipole and ∇0 denotes differentiation with respect to r₀. Put this into the Lorentz force law (Eq. 5.16) to obtain

Or, numbering the Cartesian coordinates from 1 to 3:

where i j k is the Levi-Civita symbol (+1 if i j k = 123, 231, or 312; −1 if i j k = 132, 213, or 321; 0 otherwise), in terms of which the cross-product can be written (A × B)_{i =}