$9.2.2.$ Using the formula from the previous problem, find the distribution of magnetic field induction around an infinite charged thread with a linear charge density $\rho$ if the thread moves in the longitudinal direction with a velocity $v$.
Field $E$ of an infinite charged thread $$ E = \frac{1}{2\pi\varepsilon_0 }\cdot \frac{\rho}{r} $$ then we find the induction distribution of the magnetic field induction around an infinite charged thread $$ |\vec{B}| = \varepsilon_0 \mu_0|\vec{v}\times \vec{E}| = \varepsilon_0 \mu_0 vE $$ Substituting the expression for $E$ and after transformations, we obtain the final formula for the magnetic field modulus: $$ \boxed{|\vec{B}| = \frac{\mu_0}{2\pi} \cdot \frac{\rho v}{r}} $$
$$B = \frac{\mu_0}{2\pi} \cdot \frac{\rho v}{r}$$