Magnetic induction in Relativity As we know magnetic phenomenon is a mere relativistic effect.My question is how to explain the magnetic induction in a relativistic manner?
 A: By using Coulomb law,
$$
\mathbf F = q\mathbf E = q\frac{Q\mathbf r }{|\mathbf r |^{3}},
$$
and relativistic transformation laws,
$$
\mathbf r' = \mathbf r + \Gamma \mathbf u \frac{(\mathbf u \cdot \mathbf r )}{c^{2}} - \gamma \mathbf u t , \quad \frac{\mathbf F }{\gamma \left( 1 - \frac{(\mathbf u \cdot \mathbf v)}{c^{2}}\right)} = \mathbf F ' + \gamma \frac{\mathbf v ' \cdot \mathbf F '}{c^{2}}\mathbf u  + \Gamma \mathbf u \frac{(\mathbf u \cdot \mathbf F ') }{c^{2}}, \quad \Gamma = \frac{\gamma - 1}{\frac{u^{2}}{c^{2}}},
$$
where $\mathbf u$ can be interpreted as charge $Q $ speed, $\mathbf v$ can be interpreted as test charge speed,
for $t = 0$, you will obtain an expression
$$
\mathbf F = q\mathbf E + \frac{q}{c}[\mathbf v \times \mathbf B] ,
$$
where 
$$
\mathbf E = \frac{Q\gamma \mathbf r}{\left( r^{2} + u^{2}\gamma^{2}\frac{(\mathbf u \cdot \mathbf r)^{2}}{c^{2}}\right)^{\frac{3}{2}}}, \quad \mathbf B = \frac{1}{c}[\mathbf u \times \mathbf E].
$$
As you can see according to the $\mathbf F $ and $\mathbf B$ expressions, the induction is relativistic kinematic effect which is connected with finite speed of interactions. By the other words, it can be described as delay of the electric field displacement in time. 
