$(21$ points$)$ Two very large flat conducting strings are faced parallel to each other and are separated by a small distance $d$ and are connected in such a way that the the upper string carries a uniform surface current density vec$(K)=-$Khat$(x)$ and the lower string vec$(K)=+$Khat$(x)$ flowing horizontally as shown in the figure.

a$)$ Determine in the terms of the given parameters the magnetic field vec$(B)$ between the two strings.

b$)$ Assuming that the two strings are kept at an electric potential difference $V$ determine the electric field vec$(E)(z)$ between the two plates.

c$)$ A charged particle $q$ is crossing longitudinally the space between the two strings with a constant velocity $v=-v_0$hat$(x).$ At which distance $z$ from the lower string will not be subjected to a deflection? Neglect the effect of the charge to the configuration of the fields due to the strings.

Hint: Use Ampere's law for each string separately assuming the strings have infinite size and use the superposition principle to determine the total magnetic field.