Chapter 48: Problem 2

Question: Prove that two linearly polarized light waves of equal amplitude, thier planes of vibration being at right angles to each other, cannot produce interference effects.

Solution: Let us define coordinate axes by the unit vectors ${\bf i}$ and ${\bf j}$such that ${\bf i}\cdot {\bf j}~=~0$. The two orthogonal electric vectors of equal magnitude are then defined as ${\bf E_1}~=~E {\bf i}$ and ${\bf E_2}~=~E {\bf j}$. The intensity of the sum of these two electric fields is then

$$I\propto \vert{\bf E_1}+{\bf E_2}\vert^2~=~\vert{\bf E_1}\vert^2+... ...bf E_1}\dot {\bf E_2}~=~\vert{\bf E_1}\vert^2+ \vert{\bf E_2}\vert^2~=~ I_1+I_2$$