To solve this problem, we need to find the electric field at the midpoint between two parallel infinite line charges.$\\$Given:$\\$• Two line charges with linear charge densities $+\lambda$ C/m and $-\lambda$ C/m.$\\$• The distance between the line charges is $2R. \\$The electric field due to an infinite line charge at a distance $r$ from it is given by:$\\ E = \frac{\lambda}{2 \pi \varepsilon_0 r} \\$Let's consider the midpoint between the two line charges. The distance from each line charge to the midpoint is $R. \\$The electric field due to the positive line charge at the midpoint is:$\\ E_+ = \frac{\lambda}{2 \pi \varepsilon_0 R} \\$The electric field due to the negative line charge at the midpoint is:$\\ E_- = \frac{\lambda}{2 \pi \varepsilon_0 R} \\$Since the charges are opposite, the electric fields due to each line charge at the midpoint will be in the same direction.$\\$Therefore, the net electric field at the midpoint is the sum of the magnitudes of the two fields:$\\ E_{net} = E_+ + E_- = \frac{\lambda}{2 \pi \varepsilon_0 R} + \frac{\lambda}{2 \pi \varepsilon_0 R} \\ E_{net} = \frac{2\lambda}{2 \pi \varepsilon_0 R} \\ E_{net} = \frac{\lambda}{\pi \varepsilon_0 R} \\$Thus, the electric field mid-way between the two line charges is $\frac{\lambda}{\pi \varepsilon_0 R}$ N/C.$\\$This corresponds to Option 3.