Here is a comparison between Feynman's derivation of B=vxE/c^2 and my
derivation of -div(A)=v.E/c^2 that better illustrates the similarities
between the two.
Feynman's derivation of B=vxE/c^2:
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From Feynman's "Lectures on Physics", vol. 2, pg. 26-3, where Feynman
derives B=vxE/c^2 for a charge moving with constant speed v in the
x-direction,
"For the z-component [of B],
B_z = d(A_y)/dx - d(A_x)/dy
Since A_y is zero we have just one derivative to get. Notice, however, that
A_x is just v(phi) [he has set c=1], and d/dy of v(phi) is just -v*E_y. So
B_z = v*E_y
Similarly,
B_y = d(A_x)/dz - d(A_z)/dx = +v*d(phi)/dz
and
B_y = -v*E_z
Finally, B_x is zero, since A_y and A_z are both zero. We can write the
magnetic field simply as
B = vxE [c=1]"
or with c put back in,
B = vxE/c^2
My derivation of -div(A)=v.E/c^2:
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A_x is just v_x(phi) (with c=1), A_y is v_y(phi), and A_z is v_z(phi), and
since d/dx of v_x(phi) is -v_x*E_x, d/dy of v_y(phi) is -v_y*E_y, and d/dz
of v_z(phi) is -v_z*E_z, you can write (*) as