Two infinitely long parallel line charges are $4.0\text{cm}$ apart. Line 1 carries ${\lambda }_1=+3\mu \text{C/m}$ and line 2 carries ${\lambda }_2=-3\mu \text{C/m}$. Find the magnitude of the resultant electric field at the point midway between the two lines.

Step-by-step solution

1. The distance from each line to the midpoint is half the separation: $d=2.0\text{cm}=0.020\text{m}$.
2. Electric field due to an infinite line charge: $E=\frac{\lambda }{2\pi {\epsilon }_{0}r}$, directed radially away from a positive line and toward a negative line.
3. Calculate magnitude from line 1: $E_1=\frac{3\times 10^{-6}}{2\pi (8.85\times 10^{-12})(0.020)}\approx 2.70\times 10^4\text{N/C}$, direction away from line 1.
4. Calculate magnitude from line 2: same magnitude $E_2=2.70\times 10^4\text{N/C}$, but direction toward line 2 (i.e., opposite to the field from line 1).
5. Because the lines have opposite charges, the fields at the midpoint add in the same direction (from the positive line toward the negative line).
6. Resultant field: $E=E_1+E_2=2\times 2.70\times 10^4=5.40\times 10^4\text{N/C}$, directed from line 1 toward line 2.

Concept tested: Superposition of fields from line charges

Published 19 September 2026 · How this solution was prepared and checked