6.2.10. Two infinite planes intersecting at an angle $\alpha$ divide space into four regions. What is the electric field strength in regions 1 and 2 if the surface charge density of the planes is $\pm \sigma$?
Solution
The electric field intensity for a single infinite plane with surface charge density $\sigma$ is:
$$E = \frac{\sigma}{2\epsilon_0}$$
Region 1
In the $y$-direction: $$E_{R1y} = E - E \cos{\alpha} = \frac{\sigma}{2\epsilon_0}(1-\cos{\alpha})$$
In the $x$-direction: $$E_{R1x} = E \sin{\alpha} = \frac{\sigma}{2\epsilon_0}\sin{\alpha}$$
Using $E_{R1} = \sqrt{E_{R1x}^2 + E_{R1y}^2}$ and the identity $\sin{\frac{\alpha}{2}} = \sqrt{\frac{1-\cos{\alpha}}{2}}$:
