Given Coulomb’s Law, the Law of Biot and Savart, the force equation, and the
relationship between the speed of light, permittivity, and permeability constants,
E2=q2r/(4o||r||3)
B2=oq2(v2×r)/(4r||3)
F1=q1(E2+v1×B2)
c2=1/(oo)
then
F1=q1(q2r/(4o||r||3)+v1×(oq2(v2×r)/(4r||3))) (put E2 and B2 inside F1)
=q1q2(r/o+ov1×(v2×r))/(4||r||3)
(factor common terms)
3
=q1q2(r/o+o(v2(v1•r)-r(v1•v2)))/(4||r|| )
(apply BAC-CAB identity)
=oq1q2(c2r+v2(v1•r)-r(v1•v2))/(4||r||3).
(rearrange using c2=1/(oo))
Given Scenario.gif (where v1=v2, ||v1||=v), we have (v1•r)=0 and (v1•v2)=v2, and then
F1=oq1q2r(c2-v2)/(4||r||3).
There is no relative motion since v1=v2, so there should be no magnetic component of
force. But obviously there is a magnetic component, because
lim F1=0
vic
says that something besides the Coulomb force comes into play to make the net force
zero as vic, and the derivation of the equation for F1 clearly shows that this something
else is v2 and comes from the magnetism term in the force equation.