Part 1: The edges conducting flat ribbon of resistivity $\rho = 10^{-7}\Omega \cdot m$ of thickness $t = 1$ mm, height $h = 20$ cm and length $l = 2.52$ m, are connected to a battery with a potential difference $V = 12.6$ mV.
a) Determine in terms of the given parameters the superficial current density $K$. Assume that the lines of currents are uniformly distributed on the section of the conductor, as shown in Figure 1.
Fig. 1
Part 2: The ribbon is then cut in half, and the two parts are faced with each other as shown in Figure 2. The inner edges of the two ribbons are maintained at the same potential difference of $12.6$ mV, while the outer edges are connected with another ribbon of negligible resistance.
Fig. 2
Fig. 3
Determine, in terms of the given parameters:
b) the electric field $\vec{E}(z)$ between the two plates, in terms of the given parameters,
c) the magnetic field $\vec{B}$ between the two plates. 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.
d) Assuming that the resistance is negligible which type of motion a positively charged particle $+q$ placed at rest between the ribbons halfway between the two ribbons and at $z = l/2$ will have? Describe and draw qualitatively the path that the particle will follow on figure 3.
Neglect the effect of the connecting ribbon and of the charge to the configuration of the fields between the two ribbons. Take the permeability of free space $\mu_0 = 4\pi \times 10^{-7}$ T $\cdot$ m/A.